Chapter 7, First law, internal energy Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

A mole of ideal gas is taken from a state $p_1, V_1$ to a state $p_2, V_2$ along a path forming a straight line on an indicator diagram. Find an expression for the work $\mathrm{W}$ done on the gas. Assuming the constant-volume heat capacity $C_V$ is independent of temperature, find also the internal energy change $\Delta U$ and the heat $Q$ entering the system. Apply your results to find $\Delta U, W$, and $Q$ when the initial state is at volume $21 \times 10^{-3} \mathrm{~m}^3$ and temperature $290 \mathrm{~K}$, and the final state is at $22 \times 10^{-3} \mathrm{~m}^3, 330 \mathrm{~K}$, for a gas with $\gamma=1.4$

Brandy Heflin

Numerade Educator

Problem 2

Obtain equation (7.43).

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04:28

Problem 3

By combining the ideal gas equation of state with (7.43), show that during an adiabatic expansion $T V^{\gamma-1}$ is constant and $T^\gamma p^{1-\gamma}$ is constant, when $\gamma$ is independent of temperature.

Sachin Rao

Numerade Educator

02:05

Problem 4

An ideal gas at initial pressure $p_1$ undergoes an adiabatic expansion from volume $V_1$ to volume $V_2$. Assuming $\gamma$ is constant, find the final pressure and show that the work done is
$$
W=\frac{p_1 V_1}{\gamma-1}\left(\left(\frac{V_1}{V_2}\right)^{\gamma-1}-1\right) .
$$

Manik Pulyani

Numerade Educator

01:11

Problem 5

An ideal gas is taken between the same initial and final states as in Exercise 7.1 , by an adiabatic expansion followed by heating at constant volume. Calculate the work done and heat absorbed.

Pankaj Jain

Numerade Educator

03:58

Problem 6

A container of gas is sealed by a small ball bearing which can move freely in a vertical tube. The ball is displaced vertically by a small amount and then released. Show that the period of the harmonic oscillations that result is $2 \pi(V / \gamma g A)^{1 / 2}$, where $V$ is the equilibrium volume and $A$ is the cross-sectional area of the ball (and tube). [This is the basis of Rüchardt's method to measure the adiabatic index.]

Ajay Singhal

Numerade Educator

04:16

Problem 7

Obtain equation (7.33) as follows. First, using equation (5.37), show that
$$
\frac{\mathrm{d} Q_p}{\mathrm{~d} V}=\left.C_p \frac{\partial T}{\partial V}\right|_p, \quad \frac{\mathrm{d} Q_V}{\mathrm{~d} p}=\left.C_V \frac{\partial T}{\partial p}\right|_V .
$$
Next, argue that during a change of state by $\mathrm{d} p$ and $\mathrm{d} V$, the total heat entering a $p V$ system must be
$$
\mathrm{d} Q=\frac{\mathrm{a} Q_P}{\mathrm{~d} V} \mathrm{~d} V+\frac{\mathrm{a} Q_V}{\mathrm{~d} p} \mathrm{~d} p
$$
We must proceed carefully, because we have no guarantee that this combination is a proper differential, and in fact it is not for most changes.
However, adiabatic changes have the special property that $\mathrm{d} Q=0$. Use this to obtain an expression for $(\partial p / \partial V) s$ in terms of the quantities in (7.48) and hence derive equation (7.33) or (7.34).

Khoobchandra Agrawal

Numerade Educator

08:58

Problem 8

Find expressions for $\kappa_T$ and $\kappa_S$ for an ideal gas of given $\gamma$.

Christina Bauer

Numerade Educator

01:22

Problem 9

At one atmosphere, the specific heat capacity of water at constant pressure is given to four significant figures by the formula
$$
\begin{aligned}
\frac{c_p(\theta)}{c}= & 0.996185+0.0002874\left(1+\frac{\theta}{100}\right)^{5.26} \\
& +0.011160 e^{-0.0829 \theta}
\end{aligned}
$$
where $c=4185.5 \mathrm{JK}^{-1} \mathrm{~kg}^{-1}$ and $\theta$ is the temperature in degrees Celsius. Find the heat energy required to raise the temperature of 50 grams of water from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$.

Lottie Adams

Numerade Educator

01:15

Problem 10

A spring of length $L$ obeys Hooke's law $f=k\left(L-L_0\right)$, where $f$ is the tension, $L_0$ the natural length, and $k$ is the spring constant. Find the work done on the spring when its length is changed from $L_1$ to $L_2$.

Regina Hays

Numerade Educator

02:57

Problem 11

Consider the ideal elastic substance described by equation (6.20). Show that the isothermal Young's modulus $E$ (defined as the ratio of stress to strain: $E=(\delta f / A) /\left(\delta L / L_0\right)$, where $A$ is a cross-sectional area) is given by
$$
E=\frac{K T}{A}\left(1+2 \frac{L_0^3}{L^3}\right) .
$$
Calculate the work required to stretch the substance isothermally from $L=L_0$ to $L=2 L_0$

Km Neeraj

Numerade Educator

02:10

Problem 12

[From Adkins] Below $100 \mathrm{~K}$ the specific heat capacity of diamond varies as the cube of temperature: $c_p=a T^3$. A small diamond of mass $100 \mathrm{mg}$ is cooled to $77 \mathrm{~K}$ by immersion in liquid nitrogen, and then dropped into a bath of liquid helium at its boiling point of $4.2 \mathrm{~K}$ at atmospheric pressure. In cooling the diamond, some of the helium is boiled off. The gas is collected and found to occupy a volume $2.48 \times 10^{-5} \mathrm{~m}^3$ at $0^{\circ} \mathrm{C}$ and 1 atmosphere pressure. What is the value of $a$ in the formula for the specific heat capacity of diamond? [The latent heat of vaporization of helium at $1 \mathrm{~atm}$ is $21 \mathrm{~kJ} / \mathrm{kg}$.]

Keshav Singh

Numerade Educator

01:58

Problem 13

A machine compresses 10 mole/minute of helium, modelled as an ideal gas, from 1 to $10^6$ Pa pressure. What rate of flow of cooling water, initially at $290 \mathrm{~K}$, is needed if the compression is to be made isothermal at $300 \mathrm{~K}$ ?

Dheeraj Sharma

Numerade Educator

04:14

Problem 14

A gas compressor compresses $50 \mathrm{~mol} / \mathrm{s}$ of gas adiabatically from 1 bar at $15^{\circ} \mathrm{C}$ to 10 bar. Treating the gas as ideal with adiabatic index $\gamma=1.4$, find the final temperature and the power input.

Surendra Kumar

Numerade Educator

05:23

Problem 15

[Adapted from Endem (1938)] It is desired to heat a room at initial temperature $10^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C}$. The volume of the room is $20 \mathrm{~m}^3$ and the initial pressure is 1 atm. The molar constant-volume heat capacity for air is $3.6 R$.
(i) Suppose first that there is no heat loss nor movement of air in or out of the room. Determine the heat input required to warm the room.
(ii) Now suppose air moves out of the room during the heating, such that the pressure remains constant. Show that in this case the energy of the air remaining in the room at the end is the same as that of the energy of the air in the room at the beginning.
(iii) Explain where the energy supplied as heat has gone, and calculate how much heat was supplied.

Yaqub Khan

Numerade Educator

01:51

Problem 16

(i) Explain carefully why, when gas leaks slowly out of a chamber, the expansion of the gas remaining in the chamber may be expected to be adiabatic (that is, quasistatic and without heat exchange). [Hint: choose carefully the physical system you wish to consider.]
(ii) A gas with $\gamma=5 / 3$ leaks out of a chamber. If the initial pressure is $32 p_0$ and the final pressure is $p_0$, show that the temperature falls by a factor 4 , and that $1 / 8$ of the particles remain in the chamber.

Ajay Singhal

Numerade Educator

01:20

Problem 17

In the experiment shown in Figure 7.5, a small thermometer is attached to the inside wall of the flask. The initial pressure $p_1$ is considerably larger than $p_0$. The temperature registered by this thermometer falls as the gas leaves the flask, and drops substantially below the equilibrium value given in equation (7.38), before finally rising again and settling at the equilibrium value. Explain.

Eileen Sullivan

Numerade Educator

03:09

Problem 18

A gas cylinder is mounted vertically and sealed by a freely movable light piston. The external air pressure is negligible. A mass $m_1$ is placed on the piston, and initially the system is in equilibrium, such that the weight of this mass is supported by pressure forces provided by the gas. The piston is then temporarily prevented from moving, while the mass is increased to $m_2$, and then the piston is released. Treating the gas as ideal with $\gamma=5 / 3$, find, in terms of $m_1$ and $m_2$, the ratios by which the volume and temperature of the gas change:
(i) in the case $\mathrm{A}$, where the piston moves freely and the total system energy $U_{\text {tot }}$ stays constant (where by $U_{\text {tot }}$ we mean the sum of gravitational potential energy of the mass and internal energy of the gas)
(ii) in the case B, where the piston is released gradually, such that the change in volume is quasistatic and without heat exchange.
[Ans. $V_2 / V_1=\left(3 m_1+2 m_2\right) / 5 m_2$ and $\left(m_1 / m_2\right)^{3 / 5} ; T_2 / T_1=\left(3 m_1+\right.$ $\left.2 m_2\right) / 5 m_1$ and $\left.\left(m_2 / m_1\right)^{2 / 5}\right]$ This experiment is explored further in Exercise 7.2 of chapter 17 .

Abid Hussain

Numerade Educator

22:38

Problem 19

A thermally insulated chamber contains some hot gas and a lump of metal. Initially the gas and the lump are at the same temperature $T_i$. The volume of the chamber can be changed by moving a frictionless piston. Assuming the heat capacities of the gas and the metal lump are comparable, sketch on one diagram the pressure-volume relation for the system
(a) if the pressure is reduced to atmospheric pressure po slowly enough for the temperature of the metal lump to be equal to that of the gas at all stages.
(b) if the pressure is reduced to $p_0$ fast enough for the metal lump not to $\mathrm{cool}$ at first (but the process is still quasistatic for the gas) after which the piston is further moved so as to maintain the pressure at $p_0$ until the metal lump and the gas attain the same temperature.
Use the first law to explain whether or not the final volume will be the same in these two processes. Explain which process finishes at the lower temperature. [Hint: consider the work done and use the fact that internal energy is a function of state for any given system such as the gas.]

Brandy Heflin

Numerade Educator

02:02

Problem 20

A thermally insulated and evacuated chamber is placed in a room where the pressure and temperature $T_0$ are maintained constant. Gas leaks slowly into the chamber through a small hole. Show that when the pressures are equalized, the temperature of the air in the chamber is $\gamma T_0$, where $\gamma=C_p / C_v$ and you may assume the heat capacities are independent of temperature. [Hint: imagine placing a bag around the chamber, just large enough to enclose the chamber and all the gas that finally ends up inside the chamber, and calculate the work done by the rest of the atmosphere as this bag collapses.] Consider the case of argon gas $(\gamma=5 / 3)$ at $50^{\circ} \mathrm{C}$ leaking into a flask made of tin. What happens to the flask?

Penny Riley

Numerade Educator