Usually, when studying ideal gases, it is assumed that their molar heat capacity $C_V$ at constant volume is a constant, which is independent of temperature and determined by the number of atoms in a gas molecule. However, in reality, this heat capacity can change. At very low temperatures, the thermal energy turns out to be insufficient to cause rotational motion of the molecules, and they can be considered monatomic. Therefore, the heat capacity at low temperatures turns out to be lower than at high temperatures. In this problem, we will consider several cases of variable heat capacity. At the same time, the equation of state of an ideal gas (that is, the relationship between pressure, volume, and temperature) does not change.

Part A. Heat capacity linearly dependent on temperature (5.7 points)

In this part, we will assume that over the entire temperature range of interest, the heat capacity of one mole of gas at constant volume depends on temperature as $C_{V} = C_0 + a T$ ($C_0, a > 0$ are constants, independent of temperature). All processes are carried out with one mole of gas. In all answers for this part, you may use $C_0$, $a$ and the molar gas constant $R$.

A1 Determine the dependence of the internal energy on temperature for this gas. Consider that the internal energy is zero at $T=0$.

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A2 Determine the heat capacity $C_P$ at temperature $T$ for a process at constant pressure.

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A3 Let a gas undergo an isochoric process in which the pressure increases from $p_1$ to $p_2$, and the volume is equal to $V$. Determine the amount of heat $Q_V$ transferred to the gas.

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A4 Let a gas undergo an isobaric process with pressure $p$ in which the initial volume is $V_1$ and the final volume is $V_2$. Determine the amount of heat $Q_P$ transferred to the gas.

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A5 Let's consider a cycle $123$. The pressure is proportiaonal to the volume along $1-2$, $2-3$ is isochoric, $3-1$ is isobaric. It's known that $V_2 = 2 V_1$ and heat capacity at constant volume in points $1$ and $2$ are related by the ration $C_{V2} = 1.5 C_{V1}$.

Determine a thermal efficiency of the cycle. Consider that $C_0=3R/2$.

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A6 Let a gas undergo an adiabatic process. Derive the relationship between the infinitesimal changes of volume $\mathrm{d}V$ and temperature $\mathrm{d}T$. Express the answer in terms of $C_0$, $a$, $T$, $V$ and $R$.

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A7 Using the relationship from the previous question derive the equation of adiabatic process in form
$$
f(T, V) = \mathrm{const}.
$$Function $f$ can include $C_0$, $a$ and $R$.

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A8 Let's consider a Carnot cycle $1234$. Paths $12$ and $34$ are isotermal, paths $23$ and $41$ are adiabatic. Express the volumes $V_3$, $V_4$ in terms of $V_1$, $V_2$, temperatures on the isoterms $T_+ = T_1 = T_2$, $T_- = T_3 = T_4$, and gas parameters $C_0$, $a$ and $R$.

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A9 Using a straightforward calculation, show that the thermal efficieny of the considered Carnot cycle does not deviate from that of a Carnot cycle performed with an ordinary ideal gas.

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Part B. Piecewise linear function heat capacity (3.3 points)

Let us propose a more realistic model for the heat capacity of a gas. For $T < T_1$, the heat capacity $C_V$ is constant and equal to that of monoatomic gas $C_1=3R/2$. In the range $ T_1 < T < T_2 $, the heat capcity increases linearly, reaching the value $C_2=5R/2$ (corresponds to a diatomic gas) at $T=T_2$. For $T>T_2$, the heat capacity remains constant. Let's assume that $T_2 = 2 T_1$.

The initial temperature of gas is $T_A=T_1/2$. We will increase the gas temperature to $T_B=3T_1$ via different paths.

B1 Determine the amount of heat $Q_V$ transferred to gas during isochoric heating. Express the answer in terms of $T_1$, $R$.

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B2 Determine the amount of heat $Q_P$ transferred to gas during isobaring heating. Express the answer in terms of $T_1$, $R$.

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B3 Let $V_0$ be the initial volume of gas. What will be the final volume if the temperature is increased adiabaticly? Determine the amount of work performed by the gas. Express the answers in terms of $T_1$, $R$ and $V_0$.

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