## ENGRD2210 Thermodynamics课程简介

Thermodynamics is the branch of physics that deals with the study of the relationship between heat, work, and energy. The laws of thermodynamics govern the behavior of all physical systems, including those that involve the transfer of energy as heat or work.

The First Law of Thermodynamics: The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. The first law of thermodynamics is a statement of this principle as applied to thermodynamic systems. It states that the total energy of a closed system remains constant, although it may be converted from one form to another.

The Second Law of Thermodynamics: The second law of thermodynamics states that the entropy of an isolated system never decreases over time. Entropy is a measure of the disorder or randomness of a system, and the second law of thermodynamics tells us that the natural tendency of any system is to become more disordered over time.

Thermodynamic Property Relationships: In thermodynamics, a property is a characteristic of a system that can be measured or calculated. Examples of thermodynamic properties include temperature, pressure, volume, and entropy. The relationships between these properties are described by equations of state, which relate them to each other and to the state of the system.

## PREREQUISITES

Presents the definitions, concepts, and laws of thermodynamics. Topics include the first and second laws, thermodynamic property relationships, and applications to vapor and gas power systems, refrigeration, and heat pump systems. Examples and problems are related to contemporary aspects of energy and power generation and to broader environmental issues.

Outcome 1: Students will be able to choose an appropriate system and identify interactions between system and surroundings.
Outcome 2: Obtain values of thermodynamic properties for a pure substance in a given state, using table, relations for incompressible substances, and relations for gases.

Outcome 3: Apply energy and entropy balances in the control mass (closed system) and control volume formulations to the analysis of devices and cycles.

Cornell students enroll only in ENGRD 2210. MAE 2210 for Non-CU students.

## ENGRD2210 Thermodynamics HELP（EXAM HELP， ONLINE TUTOR）

3. Consider $N$ molecules of an ideal monatomic gas, $C_V=3 N / 2$, placed in a vertical cylinder. The top of the cylinder is closed by a piston of mass $M$ and cross section $A$. Initially the piston is fixed, and the gas has volume $V_0$ and temperature $T_0$. Next, the piston is released, and after several oscillations comes to a stop. Disregarding friction and the heat capacity of the piston and cylinder, find the temperature and volume of the gas at equilibrium. The system is thermally isolated, and the pressure outside the cylinder is $P_a$.

When the piston is fixed, the gas is in equilibrium under its weight, so its pressure is $P_0 = mg/A$, where $m$ is the mass of the gas, $g$ is the acceleration due to gravity, and $A$ is the cross-sectional area of the cylinder. Since the gas is ideal and monatomic, we have $PV = Nk_BT$, where $P$ is the pressure, $V$ is the volume, $T$ is the temperature, $N$ is the number of molecules, and $k_B$ is Boltzmann’s constant.

When the piston is released, the gas expands, pushing the piston upward. The work done by the gas is $W = F\Delta h$, where $F$ is the force on the piston and $\Delta h$ is the change in height of the piston. Since the piston is moving slowly, we can assume that the pressure inside the cylinder is always in equilibrium with the external pressure $P_a$. Therefore, we have $F = P_a A$, and $\Delta h$ is the change in the position of the piston. The work done by the gas is therefore

$$W = P_a A\Delta h.$$

Since the system is thermally isolated, the total energy of the system is conserved. The initial energy of the system is the sum of the gravitational potential energy of the piston, $Mgh$, and the internal energy of the gas, $U_0 = \frac{3}{2}Nk_BT_0$. The final energy of the system is the sum of the gravitational potential energy of the piston at its new position, $Mgh’$, and the internal energy of the gas, $U_f = \frac{3}{2}Nk_BT_f$, where $h’$ is the new height of the piston.

Since the system is in equilibrium at the end, the final pressure of the gas is equal to the external pressure $P_a$. Therefore, we have

$$PV = Nk_BT = P_aV_f,$$

where $V_f$ is the final volume of the gas. Solving for $T_f$ in terms of $V_f$ gives

$$T_f = \frac{P_aV_f}{Nk_B}.$$

Conservation of energy gives

$$Mgh + U_0 = Mgh’ + U_f + W.$$

Substituting in the expressions for $U_0$, $U_f$, and $W$, and solving for $V_f$ gives

$$V_f = V_0\exp\left(\frac{Mgh}{Nk_BT_0}-\frac{Mgh’}{Nk_BT_f}-\frac{P_aA(h’-h)}{Nk_BT_f}\right).$$

Substituting in the expression for $T_f$ gives

$$V_f = V_0\exp\left(\frac{Mgh}{Nk_BT_0}-\frac{Mgh’}{Nk_BP_aV_f}-\frac{Ah’}{V_fk_B}\right).$$

This equation cannot be solved analytically, but it can be solved numerically using iterative methods. Start with an initial guess for $V_f$, and use the above equation to compute a new value of $V_f$. Repeat this process until the value of $V_f$ converges to a fixed point. Once $V_f$ is known, $T_f$ can be computed using the expression for $T_f$ above.

1. For a van der Waals gas
a) Prove that $\left(\partial C_V / \partial V\right)_T=0$.
b) Using a), determine the entropy of the monatomic gas $S(T, V)$ and its energy $E(T, V)$ to within additive constants.
Hint: In the limit $V \rightarrow \infty, C_V=3 N / 2$ for a monatomic van der Waals gas.
c) What is the final temperature when the gas is adiabatically compressed from $\left(V_1, T_1\right)$ to $V_2$. How much work is done in this compression?

a) The heat capacity at constant volume $C_V$ is given by

$$C_V = \left(\frac{\partial U}{\partial T}\right)_V,$$

where $U$ is the internal energy of the gas. For a van der Waals gas, the internal energy is a function of temperature $T$, volume $V$, and the number of molecules $N$. Taking the partial derivative of $C_V$ with respect to volume $V$ at constant temperature $T$ gives

$$\left(\frac{\partial C_V}{\partial V}\right)_T = \frac{\partial^2 U}{\partial V\partial T}.$$

Using the van der Waals equation of state,

$$\left(P+\frac{aN^2}{V^2}\right)(V-Nb) = Nk_BT,$$

we can write the internal energy as

$$U = \frac{3}{2}Nk_BT – \frac{aN^2}{V},$$

where we have added a constant of integration to ensure that $U$ goes to zero as $V$ goes to infinity. Taking the partial derivative of $U$ with respect to volume $V$ and temperature $T$, we get

$$\frac{\partial U}{\partial V} = \frac{aN^2}{V^2}, \quad \frac{\partial U}{\partial T} = \frac{3}{2}Nk_B.$$

Taking the partial derivative of $\frac{\partial U}{\partial V}$ with respect to temperature $T$ gives

$$\frac{\partial^2 U}{\partial V\partial T} = 0,$$

since $\frac{\partial}{\partial T}(\frac{aN^2}{V^2})=0$. Therefore, we have

$$\left(\frac{\partial C_V}{\partial V}\right)_T = 0.$$

b) Using the result from part (a), we can write the entropy $S$ of a van der Waals gas as

$$S(T,V) = S_0(T) + \int_{V_0}^V \frac{C_V(T)}{T}\mathrm{d}V = S_0(T) + \frac{3}{2}Nk_B\ln\left(\frac{V}{V_0}\right),$$

where $S_0(T)$ is an arbitrary function of temperature that accounts for the entropy of the gas when its volume is $V_0$, and $V_0$ is a reference volume. Note that the constant of integration is chosen so that the entropy goes to zero as $V$ goes to infinity.

Similarly, using the result from part (a) and the van der Waals equation of state, we can write the energy $E$ of the gas as

$$E(T,V) = E_0(T) + \frac{3}{2}Nk_BT – \frac{aN^2}{V},$$

where $E_0(T)$ is an arbitrary function of temperature that accounts for the energy of the gas when its volume is $V_0$. Note that the constant of integration is chosen so that the energy goes to zero as $V$ goes to infinity.

c) When the gas is adiabatically compressed from $\left(V_1, T_1\right)$ to $V_2$, we have

$$P_1V_1^{\gamma} = P_2V_2^{\gamma},$$

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.