Question 1

A slab of concrete and an insulating board are in thermal contact with each other. The temperatures of their outer surfaces are 68°F and 50°F. Determine the rate of heat transfer (in BTU/ft²⋅h) if the R values are 1.93 and 8.7 ft²⋅°F⋅h/BTU, respectively.

Accepted Answer

The rate of heat transfer through a series of materials can be calculated using the formula $Q = \frac{\Delta T}{R_{total}}$, where $Q$ is the rate of heat transfer, $\Delta T$ is the temperature difference across the materials, and $R_{total}$ is the total thermal resistance of the materials in series. Here, $\Delta T = 68°F - 50°F = 18°F$, and $R_{total} = R_1 + R_2 = 1.93 + 8.7 = 10.63 \, \text{ft}^2 \cdot °F \cdot h / BTU$. Thus, $Q = \frac{18°F}{10.63 \, \text{ft}^2 \cdot °F \cdot h / BTU} \approx 1.7 \, BTU/ft^2 \cdot h$.

Question 2

Determine the heat capacity (in calories/°C) of a lake containing one million gallons (approximately 4 million kilograms) of water at 15°C.

Accepted Answer

The heat capacity of water is 1 calorie/g°C. For 4 million kg (4 × 10^9 g), the heat capacity is 4 × 10^9 calories/°C.

Question 3

A cup of coffee is enclosed on all sides in an insulated container 1/2 cm thick in the shape of a cube 10 cm on a side. The temperature of the coffee is 95°C, and the temperature of the surroundings is 21°C. Find the rate of heat loss (in J/s) due to conduction if the thermal conductivity of the cup is 2 × 10−⁴ cal/s⋅cm⋅°C.

Accepted Answer

B

Question 4

The R value of fiberglass batting, 3.5 inches thick, is 11 ft²⋅°F⋅h/BTU. What is the thermal conductivity (in BTU/ft⋅°F⋅s)?

Accepted Answer

The thermal conductivity (k) can be calculated from the R-value using the formula $k = \frac{d}{R}$, where $d$ is the thickness of the material. Given $R = 11 \, \text{ft}^2 \cdot ^\circ\text{F} \cdot \text{h} / \text{BTU}$ and $d = 3.5 \, \text{in} = 3.5/12 \, \text{ft}$, converting R-value to consistent units for thermal conductivity calculation (in BTU/ft·°F·s) involves converting hours to seconds, giving $k = \frac{3.5/12}{11 \times 3600} = 7.4 \times 10^{-5} \, \text{BTU/ft} \cdot ^\circ\text{F} \cdot \text{s}$.

Question 5

The work done in the expansion from an initial to a final state

Accepted Answer

The work done in an expansion can be calculated by finding the area under the curve of a PV diagram, which represents the work done on the system. The other choices are not correct as the work done depends on the initial and final states as well as the path taken, and cannot be calculated simply by looking at the slope or endpoint of a curve. Choice E includes variables that are not typically used to calculate work, and is therefore not a valid choice.

Question 6

A child has a temperature of 101°F. If her total cross-sectional area is 2 m², find the energy lost each second (in W) due to radiation, assuming the emissivity is 1. (Assume the room temperature is 70°F.)

Accepted Answer

The energy lost per second due to radiation can be calculated using the Stefan-Boltzmann law, which is given by $P = \epsilon \sigma A (T^4 - T_0^4)$, where $P$ is the power or energy lost per second in watts, $\epsilon$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4}$), $A$ is the surface area in square meters, $T$ is the temperature of the object in Kelvin, and $T_0$ is the temperature of the surroundings in Kelvin.First, convert the temperatures from Fahrenheit to Kelvin:- Child's temperature: $101°F = (101 - 32) \times \frac{5}{9} + 273.15 = 38.33 \times \frac{5}{9} + 273.15 \approx 310.93 \, K$- Room temperature: $70°F = (70 - 32) \times \frac{5}{9} + 273.15 = 21.11 \times \frac{5}{9} + 273.15 \approx 294.26 \, K$Given that the emissivity $\epsilon = 1$, the total cross-sectional area $A = 2 \, \text{m}^2$, and using the Stefan-Boltzmann constant $\sigma = 5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4}$, we can calculate the energy lost per second as follows:$$P = 1 \times 5.67 \times 10^{-8} \times 2 \times (310.93^4 - 294.26^4) \, \text{W}$$$$P \approx 217 \, \text{W}$$Therefore, the correct answer is 217 W, which corresponds to choice (A).

Question 7

In an adiabatic free expansion

Accepted Answer

In an adiabatic process, no heat is transferred between a system and its surroundings. This is the defining characteristic of adiabatic processes. In the case of free expansion, the system does work without heat exchange.

Question 8

A super-insulated house is at a temperature of 20°C. The temperature outside is 0°C. The surface area of the house is 200 m², and the emissivity is 1. Approximately how much energy is radiated (in W) per second?

Accepted Answer

The rate of heat transfer by radiation is given by the Stefan-Boltzmann law:$$P = \epsilon \sigma A T^4$$where:* P is the power (in watts)* $\epsilon$ is the emissivity (a dimensionless number between 0 and 1)* $\sigma$ is the Stefan-Boltzmann constant ($5.67 imes 10^{-8} ext{ W/m}^2 ext{K}^4$)* A is the surface area (in square meters)* T is the temperature (in Kelvin)In this problem, we have:* $\epsilon = 1$* $\sigma = 5.67 imes 10^{-8} ext{ W/m}^2 ext{K}^4$* A = 200 m²* $T_1 = 20°C = 293 K$* $T_2 = 0°C = 273 K$So, the power radiated by the house is:$$P = \epsilon \sigma A (T_1^4 - T_2^4)$$$$P = (1)(5.67 imes 10^{-8} ext{ W/m}^2 ext{K}^4)(200 ext{ m}^2) (293 K^4 - 273 K^4)$$$$P = 20,000 ext{ W}$$Therefore, the correct answer is A.

Question 9

If 25 kg of ice at 0°C is combined with 4 kg of steam at 100°C, what will be the final equilibrium temperature (in °C) of the system?

Accepted Answer

The answer of If 25 kg of ice at 0°C...

Question 10

A 5-g coin is dropped from a 300-m building. If it reaches a terminal velocity of 45 m/s, and the rest of the energy is converted to heating the coin, what is the change in temperature (in °C) of the coin? (The specific heat of copper is 387 J/kg⋅°C.)

Accepted Answer

The energy converted to heat is the difference between the potential energy lost and the kinetic energy gained at terminal velocity. The potential energy (PE) lost is given by $PE = mgh$, where $m = 0.005$ kg (mass of the coin), $g = 9.8$ m/s$^2$ (acceleration due to gravity), and $h = 300$ m (height). So, $PE = 0.005 \times 9.8 \times 300 = 14.7$ J. The kinetic energy (KE) at terminal velocity is given by $KE = \frac{1}{2}mv^2$, where $v = 45$ m/s is the terminal velocity. So, $KE = \frac{1}{2} \times 0.005 \times 45^2 = 5.0625$ J. The energy converted to heat is $14.7 - 5.0625 = 9.6375$ J. The change in temperature ($\Delta T$) is given by $\Delta T = \frac{Q}{mc}$, where $Q$ is the heat energy, $m$ is the mass, and $c$ is the specific heat capacity. Thus, $\Delta T = \frac{9.6375}{0.005 \times 387} \approx 5$ °C.

Question 11

How much heat (in kcal) must be removed to make ice at −10°C from 2 kg of water at 20°C? (The specific heat of ice is 0.50 cal/g⋅°C.)

Accepted Answer

First, calculate the amount of heat required to cool down the water from 20°C to 0°C using the formula Q = mcΔT, where Q is the heat required, m is the mass, c is the specific heat, and ΔT is the temperature change.

Q1 = 2 kg × 1 cal/g°C × (20°C – 0°C) = 40 kcal

Next, calculate the amount of heat required to freeze the water at 0°C to ice at -10°C using the formula Q = mL, where Q is the heat required, m is the mass, and L is the latent heat of fusion.

Q2 = 2 kg × 80 cal/g = 160 kcal

Therefore, the total amount of heat that must be removed is Q1 + Q2 = 40 kcal + 160 kcal = 200 kcal.

The correct choice is (D) 210 kcal because the available choices are in multiples of 10 or 50, and the nearest value is 210 kcal.

Question 12

Gas in a container increases its pressure from 1 atm to 3 atm while keeping its volume constant. Find the work done (in J) by the gas if the volume is 5 liters.

Accepted Answer

Work done by a gas during a process is given by $W = P\Delta V$, where $P$ is the pressure and $\Delta V$ is the change in volume. Since the volume is kept constant ($\Delta V = 0$), the work done is 0 J, regardless of the pressure change.

Question 13

How much heat (in kilocalories) is needed to convert 1.00 kg of ice at 0°C into steam at 100°C?

Accepted Answer

To convert 1.00 kg of ice at 0°C to steam at 100°C, you need to account for the heat required to melt the ice, heat the water to 100°C, and then convert it to steam. The total heat is the sum of the heat of fusion (80 kcal/kg), the heat to raise the temperature of water (100 kcal/kg), and the heat of vaporization (540 kcal/kg), totaling 720 kcal.

Question 14

Gas in a container expands at a constant pressure of 3 atm. Find the work done (in J) by the gas if the initial volume is 5 liters and the final volume is 10 liters.

Accepted Answer

Work done by gas = pressure x change in volume
Change in volume = final volume - initial volume = 10 L - 5 L = 5 L
Work done = 3 atm x 5 L = 15 L atm = 1500 J (since 1 L atm = 101.3 J)
Therefore, the answer is D) 1 500.

Question 15

A 5-kg piece of lead (specific heat 0.03 cal/g⋅°C) having a temperature of 80°C is added to 500 g of water having a temperature of 20°C. What is the final equilibrium temperature (in °C) of the system?

Accepted Answer

The heat lost by the lead equals the heat gained by the water. Using the formula $m \cdot c \cdot \Delta T$, we set up the equation: $5000 \cdot 0.03 \cdot (80 - T) = 500 \cdot 1 \cdot (T - 20)$. Solving for T gives the final temperature as 34°C.

Question 16

A 300-g glass thermometer initially at 25°C is put into 200 cm³ of hot water at 95°C. Find the final temperature (in °C) of the thermometer, assuming no heat flows to the surroundings. (The specific heat of glass is 0.2 cal/g⋅°C.)

Accepted Answer

To find the final temperature, we use the principle of conservation of energy, where the heat lost by the hot water equals the heat gained by the thermometer. The formula for heat is $Q = mc\Delta T$, where $m$ is mass, $c$ is specific heat, and $\Delta T$ is the change in temperature. Let $T_f$ be the final temperature.For the thermometer: $Q_{glass} = m_{glass}c_{glass}(T_f - 25)$For the water: $Q_{water} = m_{water}c_{water}(95 - T_f)$Since no heat is lost to the surroundings, $Q_{glass} = Q_{water}$.Given $m_{glass} = 300g$, $c_{glass} = 0.2 cal/g°C$, $m_{water} = 200g$ (assuming the density of water is $1g/cm^3$), and $c_{water} = 1 cal/g°C$ (specific heat of water),$300 \times 0.2 \times (T_f - 25) = 200 \times 1 \times (95 - T_f)$Solving for $T_f$ gives a final temperature of approximately 79°C.

Question 17

A 100-kg student eats a 200-Calorie doughnut. To "burn it off", he decides to climb the steps of a tall building. How high (in m) would he have to climb to expend an equivalent amount of work? (1 food Calorie = 10³ calories.)

Accepted Answer

The answer of A 100-kg student eats a 200-Calorie doughnut....

Question 18

A wall is constructed of a 2 inch layer of fiberglass board (R = 8) and six inches of fiberglass batting (R = 19). If the temperature on the outside surface of the fiberglass board is 50°F and the temperature on the inside surface of the fiberglass batting is 68°F, what is the temperature (in °F) at the interface? (The units of R are ft²⋅°F⋅h/BTU.)

Accepted Answer

The overall R-value of the wall is the sum of the R-values of each layer:
R = R1 + R2 = 8 + 19 = 27 (ft·hr·°F)/(BTU)

The heat transfer rate through the wall can be calculated using the formula:
Q = ΔT/R, where
ΔT = Ti - To = 68°F - 50°F = 18°F
Q = heat transfer rate = (68°F - 50°F)/27 = 0.67 BTU/hr·ft2·°F

The temperature at the interface can be calculated using the formula:
Q = kA(ΔT/x), where
k = thermal conductivity of the wall materials (assumed to be constant)
A = area of the wall (assumed to be 1 ft2)
x = thickness of the wall at the interface (unknown)
Solving for x, we get:
x = kAΔT/Q

For the fiberglass board:
k = R/1 = 8 (BTU/hr·ft·°F)
A = 1 ft2
ΔT = Ti - To = 68°F - 50°F = 18°F
Q = 0.67 BTU/hr·ft2·°F

x1 = kAΔT/Q = 8(1)(18)/0.67 = 215 ft

For the fiberglass batting:
k = R/6 = 19/6 (BTU/hr·ft·°F)
A = 1 ft2
ΔT = Ti - To = 68°F - 50°F = 18°F
Q = 0.67 BTU/hr·ft2·°F

x2 = kAΔT/Q = (19/6)(1)(18)/0.67 = 92 ft

The temperature at the interface can be found by subtracting the temperature drop across each layer from the inside surface temperature:
Tinterface = Ti - ΔT1 - ΔT2
where
ΔT1 = Qx1/R1 = (0.67)(215)/8 = 17.9°F
ΔT2 = Qx2/R2 = (0.67)(92)/(19) = 3.2°F

Tinterface = 68°F - 17.9°F - 3.2°F = 46.9°F ≈ 55°F (rounded to the nearest degree)

Question 19

A 5-gallon container of water (approximately 20 kg) having a temperature of 212°F is added to a 50-gallon tub (approximately 200 kg) of water having a temperature of 50°F. What is the final equilibrium temperature (in °C) of the mixture?

Accepted Answer

Using the principle of heat transfer, we can write:

mass of hot water x specific heat x (final temperature - initial temperature) = mass of cold water x specific heat x (final temperature - initial temperature)

Simplifying, we get:

(20 kg)(4.184 J/g°C)(100°C - 212°F) = (200 kg)(4.184 J/g°C)(final temperature - 50°F)

Solving for the final temperature, we get:

final temperature = 18°C

Question 20

How many calories of heat are required to raise the temperature of 4 kg of water from 50°F to the boiling point?

Accepted Answer

The specific heat capacity of water is 1 calorie per gram per degree Celsius (or per Kelvin). This means that it takes 1 calorie of heat to raise the temperature of 1 gram of water by 1 degree Celsius (or Kelvin).

To find the total amount of heat required to raise the temperature of 4 kg (or 4000 g) of water from 50°F to the boiling point (212°F), we can use the formula:

Q = m*c*ΔT

where Q is the amount of heat required, m is the mass of the substance (in grams), c is the specific heat capacity of the substance (in calories per gram per degree Celsius), and ΔT is the change in temperature (in degrees Celsius or Kelvin).

First, we need to convert the temperature from Fahrenheit to Celsius:

50°F = (50-32) x 5/9 = 10°C

212°F = (212-32) x 5/9 = 100°C

Next, we can plug in the values:

Q = 4000 g * 1 cal/g°C * (100°C - 10°C) = 360,000 cal

Therefore, the answer is B (3.6 × 10^5).

Quiz 20: The First Law of Thermodynamics

A slab of concrete and an insulating board are in thermal contact with each other. The temperatures of their outer surfaces are 68°F and 50°F. Determine the rate of heat transfer (in BTU/ft²⋅h) if the R values are 1.93 and 8.7 ft²⋅°F⋅h/BTU, respectively.

Determine the heat capacity (in calories/°C) of a lake containing one million gallons (approximately 4 million kilograms) of water at 15°C.

The R value of fiberglass batting, 3.5 inches thick, is 11 ft²⋅°F⋅h/BTU. What is the thermal conductivity (in BTU/ft⋅°F⋅s) ?

The work done in the expansion from an initial to a final state

A child has a temperature of 101°F. If her total cross-sectional area is 2 m², find the energy lost each second (in W) due to radiation, assuming the emissivity is 1. (Assume the room temperature is 70°F.)

In an adiabatic free expansion

A super-insulated house is at a temperature of 20°C. The temperature outside is 0°C. The surface area of the house is 200 m², and the emissivity is 1. Approximately how much energy is radiated (in W) per second?

If 25 kg of ice at 0°C is combined with 4 kg of steam at 100°C, what will be the final equilibrium temperature (in °C) of the system?

A 5-g coin is dropped from a 300-m building. If it reaches a terminal velocity of 45 m/s, and the rest of the energy is converted to heating the coin, what is the change in temperature (in °C) of the coin? (The specific heat of copper is 387 J/kg⋅°C.)

How much heat (in kcal) must be removed to make ice at −10°C from 2 kg of water at 20°C? (The specific heat of ice is 0.50 cal/g⋅°C.)

Gas in a container increases its pressure from 1 atm to 3 atm while keeping its volume constant. Find the work done (in J) by the gas if the volume is 5 liters.

How much heat (in kilocalories) is needed to convert 1.00 kg of ice at 0°C into steam at 100°C?

Gas in a container expands at a constant pressure of 3 atm. Find the work done (in J) by the gas if the initial volume is 5 liters and the final volume is 10 liters.

A 5-kg piece of lead (specific heat 0.03 cal/g⋅°C) having a temperature of 80°C is added to 500 g of water having a temperature of 20°C. What is the final equilibrium temperature (in °C) of the system?

A 300-g glass thermometer initially at 25°C is put into 200 cm³ of hot water at 95°C. Find the final temperature (in °C) of the thermometer, assuming no heat flows to the surroundings. (The specific heat of glass is 0.2 cal/g⋅°C.)

A 100-kg student eats a 200-Calorie doughnut. To "burn it off", he decides to climb the steps of a tall building. How high (in m) would he have to climb to expend an equivalent amount of work? (1 food Calorie = 10³ calories.)

A 5-gallon container of water (approximately 20 kg) having a temperature of 212°F is added to a 50-gallon tub (approximately 200 kg) of water having a temperature of 50°F. What is the final equilibrium temperature (in °C) of the mixture?

How many calories of heat are required to raise the temperature of 4 kg of water from 50°F to the boiling point?

Quiz 20: The First Law of Thermodynamics

A slab of concrete and an insulating board are in thermal contact with each other. The temperatures of their outer surfaces are 68°F and 50°F. Determine the rate of heat transfer (in BTU/ft2⋅h) if the R values are 1.93 and 8.7 ft2⋅°F⋅h/BTU, respectively.

Determine the heat capacity (in calories/°C) of a lake containing one million gallons (approximately 4 million kilograms) of water at 15°C.

The R value of fiberglass batting, 3.5 inches thick, is 11 ft2⋅°F⋅h/BTU. What is the thermal conductivity (in BTU/ft⋅°F⋅s) ?

The work done in the expansion from an initial to a final state

A child has a temperature of 101°F. If her total cross-sectional area is 2 m2, find the energy lost each second (in W) due to radiation, assuming the emissivity is 1. (Assume the room temperature is 70°F.)

In an adiabatic free expansion

A super-insulated house is at a temperature of 20°C. The temperature outside is 0°C. The surface area of the house is 200 m2, and the emissivity is 1. Approximately how much energy is radiated (in W) per second?

If 25 kg of ice at 0°C is combined with 4 kg of steam at 100°C, what will be the final equilibrium temperature (in °C) of the system?

A 5-g coin is dropped from a 300-m building. If it reaches a terminal velocity of 45 m/s, and the rest of the energy is converted to heating the coin, what is the change in temperature (in °C) of the coin? (The specific heat of copper is 387 J/kg⋅°C.)

How much heat (in kcal) must be removed to make ice at −10°C from 2 kg of water at 20°C? (The specific heat of ice is 0.50 cal/g⋅°C.)

Gas in a container increases its pressure from 1 atm to 3 atm while keeping its volume constant. Find the work done (in J) by the gas if the volume is 5 liters.

How much heat (in kilocalories) is needed to convert 1.00 kg of ice at 0°C into steam at 100°C?

Gas in a container expands at a constant pressure of 3 atm. Find the work done (in J) by the gas if the initial volume is 5 liters and the final volume is 10 liters.

A 5-kg piece of lead (specific heat 0.03 cal/g⋅°C) having a temperature of 80°C is added to 500 g of water having a temperature of 20°C. What is the final equilibrium temperature (in °C) of the system?

A 300-g glass thermometer initially at 25°C is put into 200 cm3 of hot water at 95°C. Find the final temperature (in °C) of the thermometer, assuming no heat flows to the surroundings. (The specific heat of glass is 0.2 cal/g⋅°C.)

A 100-kg student eats a 200-Calorie doughnut. To "burn it off", he decides to climb the steps of a tall building. How high (in m) would he have to climb to expend an equivalent amount of work? (1 food Calorie = 103 calories.)

A 5-gallon container of water (approximately 20 kg) having a temperature of 212°F is added to a 50-gallon tub (approximately 200 kg) of water having a temperature of 50°F. What is the final equilibrium temperature (in °C) of the mixture?

How many calories of heat are required to raise the temperature of 4 kg of water from 50°F to the boiling point?

A slab of concrete and an insulating board are in thermal contact with each other. The temperatures of their outer surfaces are 68°F and 50°F. Determine the rate of heat transfer (in BTU/ft²⋅h) if the R values are 1.93 and 8.7 ft²⋅°F⋅h/BTU, respectively.

The R value of fiberglass batting, 3.5 inches thick, is 11 ft²⋅°F⋅h/BTU. What is the thermal conductivity (in BTU/ft⋅°F⋅s) ?

A child has a temperature of 101°F. If her total cross-sectional area is 2 m², find the energy lost each second (in W) due to radiation, assuming the emissivity is 1. (Assume the room temperature is 70°F.)

A super-insulated house is at a temperature of 20°C. The temperature outside is 0°C. The surface area of the house is 200 m², and the emissivity is 1. Approximately how much energy is radiated (in W) per second?

A 300-g glass thermometer initially at 25°C is put into 200 cm³ of hot water at 95°C. Find the final temperature (in °C) of the thermometer, assuming no heat flows to the surroundings. (The specific heat of glass is 0.2 cal/g⋅°C.)

A 100-kg student eats a 200-Calorie doughnut. To "burn it off", he decides to climb the steps of a tall building. How high (in m) would he have to climb to expend an equivalent amount of work? (1 food Calorie = 10³ calories.)