A sealed rigitd tank contains 30 moles of an ideal gas, at an initial temperature of $270 \mathrm {~K}$ The pressure of the gas is increased until the final pressure equals 1.40 times the initial pressure. The heat capacity at constant pressure of the gas is $32.0 \mathrm {~J} / \mathrm { mol } \cdot \mathrm { K }$ What is the change in the internal (thermal) energy of the gas during this process? (R = 8.31 J/mol ∙ K)
A) 77 kJ
B) 100 kJ
C) 130 kJ
D) -50 kJ
E) -23 kJ

Question

Quizplus · Accepted Answer

We can use the First Law of Thermodynamics:
$$\Delta U = Q - W$$
where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done on the system.
Since the tank is rigid, no work is done ($W = 0$). Therefore, we only need to calculate the heat added to the system:
$$Q = n C_p \Delta T$$
where $n$ is the number of moles, $C_p$ is the heat capacity at constant pressure, and $\Delta T$ is the change in temperature.
Since the gas is ideal, we can use the Ideal Gas Law:
$$PV = nRT$$
where $P$ is the pressure, $V$ is the volume, $R$ is the gas constant, and $T$ is the temperature.
The initial conditions are: $n=30$, $T_1 = 270\mathrm{K}$, $P_1=P$, and $V_1 = \frac{nRT_1}{P_1}$.
The final conditions are: $n=30$, $T_2 = T_1 + \Delta T$, $P_2 = 1.4P_1$, and $V_2 = \frac{nRT_2}{P_2}$.
We can eliminate the volume terms by taking the ratio of the ideal gas laws at the initial and final states:
$$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$$
Substituting for $V_1$ and $V_2$, and solving for $\Delta T$, we get:
$$\Delta T = \frac{P_2-P_1}{nR} \left(\frac{nRT_1}{P_1} ight) = \frac{P_2}{P_1}\frac{R}{C_p}T_1 - T_1$$
Plugging in the values from the problem, we get:
$$\Delta T = 78.8\mathrm{K}$$
Finally, plugging in the values for $n$, $C_p$, and $\Delta T$ into the expression for $Q$, we get:
$$Q = nC_p\Delta T = 75.8\mathrm{kJ}$$
Therefore, the change in internal energy is:
$$\Delta U = Q - W = 75.8\mathrm{kJ} - 0 = 75.8\mathrm{kJ} \approx \boxed{	extbf{(A) }77	ext{ kJ}}$$

A Sealed Rigitd Tank Contains 30 Moles of an Ideal 270 K270 \mathrm {~K}270 K

A Sealed Rigitd Tank Contains 30 Moles of an Ideal $270 \mathrm {~K}$