## Calculation of $\frac{dT}{dp}$ for water

• Clausius-Clapeyron
• assignment Vapor pressure equation

assignment Homework

##### Vapor pressure equation
phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that $\Delta V \approx V_g$. Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.
1. Solve for $\frac{dp}{dT}$ in terms of the pressure of the vapor and the latent heat $L$ and the temperature.

2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

• face Phase transformations

face Lecture

120 min.

##### Phase transformations
Thermal and Statistical Physics 2020

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
• assignment Heat of vaporization of ice

assignment Homework

##### Heat of vaporization of ice
Vaporization Heat Thermal and Statistical Physics 2020 The pressure of water vapor over ice is 518 Pa at $-2^\circ\text{C}$. The vapor pressure of water at its triple point is 611 Pa, at 0.01$^\circ\text{C}$ (see Estimate in $\text{J mol}^{-1}$ the heat of vaporization of ice just under freezing. How does this compare with the heat of vaporization of water?
• group Squishability'' of Water Vapor (Contour Map)

group Small Group Activity

30 min.

##### “Squishability” of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking $-\partial V/\partial P$ holding either temperature or entropy fixed.
• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (3 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• assignment Basic Calculus: Practice Exercises

assignment Homework

##### Basic Calculus: Practice Exercises
Static Fields 2022 (3 years) Determine the following derivatives and evaluate the following integrals.
1. $\frac{d}{du}\left(u^2\sin u\right)$
2. $\frac{d}{dz}\left(\ln(z^2+1)\right)$
3. $\displaystyle\int v\cos(v^2)\,dv$
4. $\displaystyle\int v\cos v\,dv$
• group Heat and Temperature of Water Vapor (Remote)

group Small Group Activity

5 min.

##### Heat and Temperature of Water Vapor (Remote)

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

assignment Homework

ideal gas internal energy engine Energy and Entropy 2020

A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation $pV = Nk_BT$. You may also use the property of an ideal gas that the internal energy depends only on the temperature $T$, i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by $C_V=\frac52Nk_B$, which means the internal energy is given by $U=\frac52Nk_BT$.

Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

1. If the air is initially at room temperature (taken as $20^{o}C$) and is then compressed adiabatically to $\frac1{15}$ of the original volume, what final temperature is attained (before fuel injection)?

2. By what factor does the pressure increase?

• group Quantifying Change

group Small Group Activity

30 min.

##### Quantifying Change

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
• Thermal and Statistical Physics 2020 Calculate based on the Clausius-Clapeyron equation the value of $\frac{dT}{dp}$ near $p=1\text{atm}$ for the liquid-vapor equilibrium of water. The heat of vaporization at $100^\circ\text{C}$ is $2260\text{ J g}^{-1}$. Express the result in kelvin/atm.