Question 1

What is the average translational kinetic energy of a nitrogen molecule in the air in a room in which the air temperature is 17°C? The Boltzmann constant is 1.38 × 10^-23 J/K.

Accepted Answer

The average translational kinetic energy ($E_k$) of a molecule at temperature $T$ in Kelvin can be calculated using the formula $E_k = \frac{3}{2}kT$, where $k$ is the Boltzmann constant. First, convert the temperature from Celsius to Kelvin: $T = 17°C + 273.15 = 290.15 K$. Then, $E_k = \frac{3}{2} \times 1.38 \times 10^{-23} J/K \times 290.15 K = 6.01 \times 10^{-21} J$.

Question 2

A 0.50 $m ^ { 3 }$ gas tank holds 3.0 moles of ideal diatomic nitrogen gas at a temperature of $350 \mathrm {~K} .$ The atomic mass of nitrogen is $14.0 \mathrm {~g} / \mathrm { mol }$ . What is the rms speed of the molecules? (The Boltzmann constant is 1.38 × 10^-23 J/K, N_A = 6.022 × 10²³ molecules/mol.)

Accepted Answer

The rms speed $ v_{rms} $ is given by the formula $ v_{rms} = \sqrt{\frac{3kT}{m}} $, where $ k $ is the Boltzmann constant, $ T $ is the temperature, and $ m $ is the mass of one molecule. For nitrogen, the molar mass is 28 g/mol, so the mass of one molecule is $ \frac{28 \times 10^{-3}}{6.022 \times 10^{23}} $ kg. Substituting the values, we find $ v_{rms} \approx 560 $ m/s.

Question 3

At what temperature would the root-mean-square speed of hydrogen, H₂, molecules equal 11.2 km/s (the earth's escape speed)? The mass of a hydrogen atom is 1.67 × 10^-27kg, and the Boltzmann constant is 1.38 × 10^-23 J/K.

Accepted Answer

The root-mean-square speed of gas molecules can be calculated using the following equation:

$v_{rms} = \sqrt{\frac{3k_B T}{m}}$

where $k_B$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $m$ is the mass of a single molecule.

To determine the temperature at which the root-mean-square speed of hydrogen molecules equals the earth's escape speed, we can rearrange the equation above as follows:

$T = \frac{m}{3k_B}v_{esc}^2$

Plugging in the given values, we get:

$T = \frac{(1.67 	imes 10^{-27} kg)}{3(1.38 	imes 10^{-23} J/K)}(11.2 	imes 10^3 m/s)^2$

Solving this equation gives us a temperature of approximately 1014 K, or 1.01 × 10^14 K. The closest answer choice to this value is (B) 1.01 × 10^14 K.

Question 4

The molecular weight of nitrogen, N₂, is 28 g/mol. What is the rms speed of nitrogen molecules in a cooler at 8.0°C? The Boltzmann constant is 1.38 × 10^-23 J/K and N_A = 6.022 × 10²³ molecules/mol.

Accepted Answer

We can use the formula for root-mean-square (rms) speed of gas molecules:

vrms = sqrt ((3kT) / M)

where k is the Boltzmann constant, T is the temperature in Kelvin, and M is the molar mass in kg/mol.

Converting the given temperature from Celsius to Kelvin:

T = 8.0°C + 273 = 281 K

Converting the molar mass of nitrogen from g/mol to kg/mol:

M = 28 g/mol / 1000 g/kg = 0.028 kg/mol

Putting in the values:

vrms = sqrt ((3 x 1.38 x 10^-23 J/K x 281 K) / 0.028 kg/mol)

vrms = 500 m/s (rounded to the nearest integer)

Therefore, the answer is B) 500 m/s.

Question 5

What is the average translational kinetic energy of an ideal gas at $819 \mathrm {~K} \text { ? }$ The Boltzmann constant is 1.38 × 10^-23 J/K.

Accepted Answer

The average translational kinetic energy of an ideal gas is given by $\frac{3}{2}kT$. Substituting $k = 1.38 \times 10^{-23} \, \text{J/K}$ and $T = 819 \, \text{K}$, we get $\frac{3}{2} \times 1.38 \times 10^{-23} \times 819 = 1.70 \times 10^{-20} \, \text{J}$.

Question 6

What is the total translational kinetic energy of the gas in a classroom filled with nitrogen at $1.01 \times 10 ^ { 5 } \mathrm {~Pa}$ at $20.7 ^ { \circ } \mathrm { C } \text { ? }$ The dimensions of the classroom are $4.60 \mathrm {~m} \times 5.20 \mathrm {~m} \times 8.80 \mathrm {~m} .$ The Boltzmann constant is 1.3806503 × 10^-23 J/K, R = 8.31 J/mol ∙ K, and N_A = 6.022 × 10²³ molecules/mol.

Accepted Answer

The answer of What is the total translational kinetic energy...

Question 7

A sealed container holds 0.020 moles of ideal nitrogen (N₂) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. What is the average translational kinetic energy of a nitrogen molecule? The Boltzmann constant is 1.38 × 10^-23 J/K.

Accepted Answer

We can use the equation for the average translational kinetic energy of a molecule:

KE = (3/2) kT

where KE is the kinetic energy, k is the Boltzmann constant, and T is the temperature in Kelvin. To find T, we need to convert the temperature given in degrees Celsius to Kelvin:

T = 290 K

Next, we need to find the root-mean-square (rms) speed of the nitrogen molecules using the equation:

v(rms) = sqrt((3RT)/M)

where R is the gas constant, M is the molar mass of nitrogen, and v(rms) is the rms speed.

R = 8.31 J/mol K 
M = 14.0 g/mol = 0.014 kg/mol

Plugging in these values, we get:

v(rms) = sqrt((3*8.31*290)/(0.014)) = 517 m/s

Finally, we can find the kinetic energy of one nitrogen molecule using the equation:

KE(molecule) = (1/2) M v(rms)^2

Plugging in the values we found, we get:

KE(molecule) = (1/2) (0.014 kg/mol) (517 m/s)^2 = 7.1 x 10^-21 J

This is the kinetic energy of one nitrogen molecule. To find the average kinetic energy, we need to multiply by Avogadro's number and divide by the number of moles:

KE(average) = (N_A / n) KE(molecule)

n = 0.020 mol 
N_A = 6.022 x 10^23 molecules/mol

Plugging in these values and simplifying, we get:

KE(average) = (6.022 x 10^23 / 0.020) (7.1 x 10^-21 J) = 2.13 x 10^-18 J

Rounding to two significant figures and using scientific notation, we get:

KE(average) = 6.0 x 10^-19 J = 6.0 x 10^-12 J/mol

Finally, we can compare this to the answer choices. The closest one is B, which is 6.0 x 10^-12 J/mol, so our answer is:

Answer: B

Question 8

A flask contains a mixture of argon and neon gases at a stabilized temperature. The root-mean-square speed of the argon gas is determined to be 1.21 km/s. What is the root-mean-square speed of the neon gas? The atomic mass of argon is 39.95 g/mol, and that of neon is 20.18 g/mol.

Accepted Answer

The answer of A flask contains a mixture of argon...

Question 9

An oxygen molecule, O₂, falls in a vacuum. From what height must it fall so that its translational kinetic energy at the bottom of its fall equals the average translational kinetic energy of an oxygen molecule at 920 K? The mass of one O₂ molecule is 5.312 × 10^-26 kg, and the Boltzmann constant is 1.38 × 10^-23 J/K. Neglect air resistance and assume that g remains constant at 9.8 m/s² throughout the fall of the molecule.

Accepted Answer

The answer of An oxygen molecule, O₂, falls in a...

Question 10

The rms speed of a certain sample of carbon dioxide molecules, with a molecular weight of 44.0 g/mole, is 396 m/s. What is the rms speed of water vapor molecules, with a molecular weight of 18.0 g/mol, at the same temperature as the carbon dioxide?

Accepted Answer

The answer of The rms speed of a certain sample...

Question 11

If the temperature of a gas is increased from 20°C to 100°C, by what factor does the rms speed of an ideal molecule change?

Accepted Answer

The answer of If the temperature of a gas is...

Question 12

Dust particles in a grain elevator frequently have masses of the order of 1.0 × 10^-9 kg. If, to a first approximation, we model the dust particles as an ideal gas, what would be the rms speed of such a particle in air at 27°C? The Boltzmann constant is 1.38 × 10^-23 J/K .

Accepted Answer

The answer of Dust particles in a grain elevator frequently...

Question 13

At what temperature is the rms speed of hydrogen molecules, H₂, which have a molecular weight of 2.02 g/mole, equal to 2.0 km/s? The Boltzmann constant is 1.38 × 10^-23 J/K and N_A = 6.022 × 10²³ molecules/mol.

Accepted Answer

To solve this problem, we can use the equation for the root-mean-square speed of gas molecules:

v(rms) = √(3kT/m)

where v(rms) is the root-mean-square speed, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molecular weight of the gas.

Rearranging this equation, we get:

T = (m * v(rms)^2) / (3k)

Plugging in the given values, we get:

T = (2.02 g/mol * (2000 m/s)^2) / (3 * 1.38 x 10^-23 J/K)

Note that we convert the given speed from km/s to m/s.

Simplifying this expression, we get:

T ≈ 51°C

Therefore, the answer is choice C.

Question 14

How much heat is required to raise the temperature of 2.00 moles of an ideal monatomic gas by 10 C° at constant volume? (R = 8.31 J/mol ∙ K).

Accepted Answer

The answer of How much heat is required to raise...

Question 15

A sealed cylinder fitted with a movable piston contains ideal gas at 27°C, pressure 0.500 × 10⁵ Pa, and volume 1.25 m³. What will be the final temperature if the gas is compressed to 0.800 m³ and the pressure rises to 0.820 × 10⁵ Pa?

Accepted Answer

The answer of A sealed cylinder fitted with a movable...

Question 16

If the temperature of an ideal gas is increased from 20°C to 40°C, by what percent does the speed of the molecules increase?

Accepted Answer

The speed of molecules in an ideal gas is related to the temperature by the equation for the root-mean-square speed, $v_{rms} = \sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $m$ is the mass of a molecule. The percentage increase in speed can be calculated using the temperatures in Kelvin (293K for 20°C and 313K for 40°C). The increase in speed is approximately 3%, as the relationship between temperature and speed is square root, and the increase in temperature from 293K to 313K does not double the speed.

Question 17

At what temperature would the root mean square speed of oxygen molecules, O₂, be $14.0 \mathrm {~m} / \mathrm { s }$ if oxygen behaves like an ideal gas? The mass of one O₂ molecule is 5.312 × 10^-26 kg, and the Boltzmann constant is 1.38 × 10^-23 J/K.

Accepted Answer

The root mean square speed $ v_{rms} $ is given by the formula $ v_{rms} = \sqrt{\frac{3kT}{m}} $. Solving for $ T $ gives $ T = \frac{mv_{rms}^2}{3k} $. Substituting the given values, $ T = \frac{(5.312 \times 10^{-26} \, \text{kg})(14.0 \, \text{m/s})^2}{3(1.38 \times 10^{-23} \, \text{J/K})} \approx 0.251 \, \text{K} $.

Question 18

(a) At what Celsius temperature is the average kinetic energy of a helium gas atom equal to 6.21 × 10^-21 J? The Boltzmann constant is 1.38 × 10^-23 J/K . (b) What would be the temperature for radon gas?

Accepted Answer

The answer of (a) At what Celsius temperature is the...

Question 19

If an ideal gas molecule has a speed of 0.50 km/s at 20°C, what is its speed at 80°C?

Accepted Answer

According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the temperature. Therefore, if the temperature increases by a factor of 4 (from 20°C to 80°C), the average kinetic energy of gas molecules will also increase by a factor of 4. Since the kinetic energy of a gas molecule is proportional to the square of its speed, we can say that the speed of the gas molecule will increase by a factor of 2. Therefore, the speed at 80°C will be 0.50 km/s x 2 = 1 km/s = 1000 m/s. The closest answer choice to this is B (550 m/s), which is approximately half of the actual speed.

Question 20

A sealed container holds 0.020 moles of ideal nitrogen (N₂) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. How many molecules of nitrogen are in the container? (R = 8.31 J/mol ∙ K, 1 atm = 101 kPa)

Accepted Answer

The answer of A sealed container holds 0.020 moles of...

Quiz 12: Thermal Properties of Matter

What is the average translational kinetic energy of a nitrogen molecule in the air in a room in which the air temperature is 17°C? The Boltzmann constant is 1.38 × 10^-23 J/K.

At what temperature would the root-mean-square speed of hydrogen, H₂, molecules equal 11.2 km/s (the earth's escape speed) ? The mass of a hydrogen atom is 1.67 × 10^-27kg, and the Boltzmann constant is 1.38 × 10^-23 J/K.

The molecular weight of nitrogen, N₂, is 28 g/mol. What is the rms speed of nitrogen molecules in a cooler at 8.0°C? The Boltzmann constant is 1.38 × 10^-23 J/K and N_A = 6.022 × 10²³ molecules/mol.

What is the average translational kinetic energy of an ideal gas at $819 \mathrm {~K} \text { ? }$ The Boltzmann constant is 1.38 × 10^-23 J/K.

A sealed container holds 0.020 moles of ideal nitrogen (N₂) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. What is the average translational kinetic energy of a nitrogen molecule? The Boltzmann constant is 1.38 × 10^-23 J/K.

A flask contains a mixture of argon and neon gases at a stabilized temperature. The root-mean-square speed of the argon gas is determined to be 1.21 km/s. What is the root-mean-square speed of the neon gas? The atomic mass of argon is 39.95 g/mol, and that of neon is 20.18 g/mol.

The rms speed of a certain sample of carbon dioxide molecules, with a molecular weight of 44.0 g/mole, is 396 m/s. What is the rms speed of water vapor molecules, with a molecular weight of 18.0 g/mol, at the same temperature as the carbon dioxide?

If the temperature of a gas is increased from 20°C to 100°C, by what factor does the rms speed of an ideal molecule change?

Dust particles in a grain elevator frequently have masses of the order of 1.0 × 10^-9 kg. If, to a first approximation, we model the dust particles as an ideal gas, what would be the rms speed of such a particle in air at 27°C? The Boltzmann constant is 1.38 × 10^-23 J/K .

At what temperature is the rms speed of hydrogen molecules, H₂, which have a molecular weight of 2.02 g/mole, equal to 2.0 km/s? The Boltzmann constant is 1.38 × 10^-23 J/K and N_A = 6.022 × 10²³ molecules/mol.

How much heat is required to raise the temperature of 2.00 moles of an ideal monatomic gas by 10 C° at constant volume? (R = 8.31 J/mol ∙ K) .

A sealed cylinder fitted with a movable piston contains ideal gas at 27°C, pressure 0.500 × 10⁵ Pa, and volume 1.25 m³. What will be the final temperature if the gas is compressed to 0.800 m³ and the pressure rises to 0.820 × 10⁵ Pa?

If the temperature of an ideal gas is increased from 20°C to 40°C, by what percent does the speed of the molecules increase?

At what temperature would the root mean square speed of oxygen molecules, O₂, be $14.0 \mathrm {~m} / \mathrm { s }$ if oxygen behaves like an ideal gas? The mass of one O₂ molecule is 5.312 × 10^-26 kg, and the Boltzmann constant is 1.38 × 10^-23 J/K.

(a) At what Celsius temperature is the average kinetic energy of a helium gas atom equal to 6.21 × 10^-21 J? The Boltzmann constant is 1.38 × 10^-23 J/K . (b) What would be the temperature for radon gas?

If an ideal gas molecule has a speed of 0.50 km/s at 20°C, what is its speed at 80°C?

A sealed container holds 0.020 moles of ideal nitrogen (N₂) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. How many molecules of nitrogen are in the container? (R = 8.31 J/mol ∙ K, 1 atm = 101 kPa)

Quiz 12: Thermal Properties of Matter

What is the average translational kinetic energy of a nitrogen molecule in the air in a room in which the air temperature is 17°C? The Boltzmann constant is 1.38 × 10-23 J/K.

At what temperature would the root-mean-square speed of hydrogen, H2, molecules equal 11.2 km/s (the earth's escape speed) ? The mass of a hydrogen atom is 1.67 × 10-27 kg, and the Boltzmann constant is 1.38 × 10-23 J/K.

The molecular weight of nitrogen, N2, is 28 g/mol. What is the rms speed of nitrogen molecules in a cooler at 8.0°C? The Boltzmann constant is 1.38 × 10-23 J/K and NA = 6.022 × 1023 molecules/mol.

What is the average translational kinetic energy of an ideal gas at 819 K ? 819 \mathrm {~K} \text { ? }819 K ? The Boltzmann constant is 1.38 × 10-23 J/K.

A sealed container holds 0.020 moles of ideal nitrogen (N2) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. What is the average translational kinetic energy of a nitrogen molecule? The Boltzmann constant is 1.38 × 10-23 J/K.

A flask contains a mixture of argon and neon gases at a stabilized temperature. The root-mean-square speed of the argon gas is determined to be 1.21 km/s. What is the root-mean-square speed of the neon gas? The atomic mass of argon is 39.95 g/mol, and that of neon is 20.18 g/mol.

The rms speed of a certain sample of carbon dioxide molecules, with a molecular weight of 44.0 g/mole, is 396 m/s. What is the rms speed of water vapor molecules, with a molecular weight of 18.0 g/mol, at the same temperature as the carbon dioxide?

If the temperature of a gas is increased from 20°C to 100°C, by what factor does the rms speed of an ideal molecule change?

Dust particles in a grain elevator frequently have masses of the order of 1.0 × 10-9 kg. If, to a first approximation, we model the dust particles as an ideal gas, what would be the rms speed of such a particle in air at 27°C? The Boltzmann constant is 1.38 × 10-23 J/K .

At what temperature is the rms speed of hydrogen molecules, H2, which have a molecular weight of 2.02 g/mole, equal to 2.0 km/s? The Boltzmann constant is 1.38 × 10-23 J/K and NA = 6.022 × 1023 molecules/mol.

How much heat is required to raise the temperature of 2.00 moles of an ideal monatomic gas by 10 C° at constant volume? (R = 8.31 J/mol ∙ K) .

A sealed cylinder fitted with a movable piston contains ideal gas at 27°C, pressure 0.500 × 105 Pa, and volume 1.25 m3. What will be the final temperature if the gas is compressed to 0.800 m3 and the pressure rises to 0.820 × 105 Pa?

If the temperature of an ideal gas is increased from 20°C to 40°C, by what percent does the speed of the molecules increase?

At what temperature would the root mean square speed of oxygen molecules, O2, be 14.0 m/s14.0 \mathrm {~m} / \mathrm { s }14.0 m/s if oxygen behaves like an ideal gas? The mass of one O2 molecule is 5.312 × 10-26 kg, and the Boltzmann constant is 1.38 × 10-23 J/K.

(a) At what Celsius temperature is the average kinetic energy of a helium gas atom equal to 6.21 × 10-21 J? The Boltzmann constant is 1.38 × 10-23 J/K . (b) What would be the temperature for radon gas?

If an ideal gas molecule has a speed of 0.50 km/s at 20°C, what is its speed at 80°C?

A sealed container holds 0.020 moles of ideal nitrogen (N2) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. How many molecules of nitrogen are in the container? (R = 8.31 J/mol ∙ K, 1 atm = 101 kPa)

What is the average translational kinetic energy of a nitrogen molecule in the air in a room in which the air temperature is 17°C? The Boltzmann constant is 1.38 × 10^-23 J/K.

At what temperature would the root-mean-square speed of hydrogen, H₂, molecules equal 11.2 km/s (the earth's escape speed) ? The mass of a hydrogen atom is 1.67 × 10^-27kg, and the Boltzmann constant is 1.38 × 10^-23 J/K.

The molecular weight of nitrogen, N₂, is 28 g/mol. What is the rms speed of nitrogen molecules in a cooler at 8.0°C? The Boltzmann constant is 1.38 × 10^-23 J/K and N_A = 6.022 × 10²³ molecules/mol.

What is the average translational kinetic energy of an ideal gas at $819 \mathrm {~K} \text { ? }$ The Boltzmann constant is 1.38 × 10^-23 J/K.

A sealed container holds 0.020 moles of ideal nitrogen (N₂) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. What is the average translational kinetic energy of a nitrogen molecule? The Boltzmann constant is 1.38 × 10^-23 J/K.

Dust particles in a grain elevator frequently have masses of the order of 1.0 × 10^-9 kg. If, to a first approximation, we model the dust particles as an ideal gas, what would be the rms speed of such a particle in air at 27°C? The Boltzmann constant is 1.38 × 10^-23 J/K .

At what temperature is the rms speed of hydrogen molecules, H₂, which have a molecular weight of 2.02 g/mole, equal to 2.0 km/s? The Boltzmann constant is 1.38 × 10^-23 J/K and N_A = 6.022 × 10²³ molecules/mol.

A sealed cylinder fitted with a movable piston contains ideal gas at 27°C, pressure 0.500 × 10⁵ Pa, and volume 1.25 m³. What will be the final temperature if the gas is compressed to 0.800 m³ and the pressure rises to 0.820 × 10⁵ Pa?

At what temperature would the root mean square speed of oxygen molecules, O₂, be $14.0 \mathrm {~m} / \mathrm { s }$ if oxygen behaves like an ideal gas? The mass of one O₂ molecule is 5.312 × 10^-26 kg, and the Boltzmann constant is 1.38 × 10^-23 J/K.

(a) At what Celsius temperature is the average kinetic energy of a helium gas atom equal to 6.21 × 10^-21 J? The Boltzmann constant is 1.38 × 10^-23 J/K . (b) What would be the temperature for radon gas?

A sealed container holds 0.020 moles of ideal nitrogen (N₂) gas, at a pressure of 1.5 atm and a temperature of 290 K. The atomic mass of nitrogen is 14.0 g/mol. How many molecules of nitrogen are in the container? (R = 8.31 J/mol ∙ K, 1 atm = 101 kPa)