Chapter 3: The First Law of ThermodynamicsContent check — University Physics 2, Chapter 3: The First Law of Thermodynamics

How much work is done by the gas, as given in Figure 3.6, when it expands quasi-statically along the path ADC?

The quantities below represent four different transitions between the same initial and final state. Fill in the blanks.

Why was it necessary to state that the process of Example 3.5 is quasi-static?

When 1.00 g of ammonia boils at atmospheric pressure and $\mathrm{-33.0}\text{°}\text{C,}$ its volume changes from 1.47 to $1130{\text{cm}}^3$. Its heat of vaporization at this pressure is $1.37\times {10}^6\text{J/kg}\text{.}$ What is the change in the internal energy of the ammonia when it vaporizes?

Consider these scenarios and state whether work is done by the system on the environment (SE) or by the environment on the system (ES): (a) opening a carbonated beverage; (b) filling a flat tire; (c) a sealed empty gas can expands on a hot day, bowing out the walls.

When a liquid is vaporized, its change in internal energy is not equal to the heat added. Why?

Is it possible for the temperature of a system to remain constant when heat flows into or out of it? If so, give examples.

Does adding heat to a system always increase its internal energy?

When a gas expands isothermally, it does work. What is the source of energy needed to do this work?

It is unlikely that a process can be isothermal unless it is a very slow process. Explain why. Is the same true for isobaric and isochoric processes? Explain your answer.

Most materials expand when heated. One notable exception is water between $0\text{°}\text{C}$ and $4\text{°}\text{C,}$ which actually decreases in volume with the increase in temperature. Which is greater for water in this temperature region, $C_p$ or $C_V$ ?

Is it possible for $\text{γ}$ to be smaller than unity?

There is no change in the internal energy of an ideal gas undergoing an isothermal process since the internal energy depends only on the temperature. Is it therefore correct to say that an isothermal process is the same as an adiabatic process for an ideal gas? Explain your answer.

A gas follows $pV=bp+c_T$ on an isothermal curve, where p is the pressure, V is the volume, b is a constant, and c is a function of temperature. If c can be determined from the measurement of a single other thermodynamic parameter, it establishes a temperature scale for this gas. Show that a temperature scale under an isochoric process can be established with this gas, with pressure being the measured parameter, and is identical to that of an ideal gas.

Find the isobaric expansion coefficient $dV\text{/}dT$ and the isothermal pressure-volume coefficient $dV\text{/}dP$ of a solid that has equation of state $V+bpT–c{T}^2=0$.

It takes 500 J of work to compress quasi-statically 0.50 mol of an ideal gas to one-fifth its original volume. Calculate the temperature of the gas, assuming it remains constant during the compression.

In a quasi-static isobaric expansion, 500 J of work are done by the gas. If the gas pressure is 0.80 atm, what is the fractional increase in the volume of the gas, assuming it was originally at 20.0 L?

An ideal gas expands quasi-statically and isothermally from a state with pressure p and volume V to a state with volume 4V. Show that the work done by the gas in the expansion is pV(ln 4).

(a) Calculate the work done on the gas along the closed path shown below. The curved section between R and S is semicircular when plotted using the indicated unit. (b) If the process is carried out in the opposite direction, what is the work done on the gas?

A dilute gas at a pressure of 2.0 atm and a volume of 4.0 L is taken through the following quasi-static steps: (a) an isobaric expansion to a volume of 10.0 L, (b) an isochoric change to a pressure of 0.50 atm, (c) an isobaric compression to a volume of 4.0 L, and (d) an isochoric change to a pressure of 2.0 atm. Show these steps on a pV diagram and determine from your graph the net work done by the gas.

What is the internal energy of 6.00 mol of an ideal monatomic gas at $200\text{°}\text{C}$ ?

Two monatomic ideal gases A and B are at the same temperature. If 1.0 g of gas A has the same internal energy as 0.10 g of gas B, what are (a) the ratio of the number of moles of each gas and (b) the ratio of the atomic masses of the two gases?

Find the work done in the quasi-static processes shown below. The states are given as (p, V) values for the points in the pV plane: 1 (3 atm, 4 L), 2 (3 atm, 6 L), 3 (5 atm, 4 L), 4 (2 atm, 6 L), 5 (4 atm, 2 L), 6 (5 atm, 5 L), and 7 (2 atm, 5 L).

In an expansion of gas, 500 J of work are done by the gas. If the internal energy of the gas increased by 80 J in the expansion, how much heat does the gas absorb?

As shown below, if the heat absorbed by the gas along AB is 400 J, determine the quantities of heat absorbed along (a) ADB; (b) ACB; and (c) ADCB.

(a) What is the change in internal energy for the process represented by the closed path shown below? (b) How much heat is exchanged? (c) If the path is traversed in the opposite direction, how much heat is exchanged?

When a gas expands along AB (see below), it does 20 J of work and absorbs 30 J of heat. When the gas expands along AC, it does 40 J of work and absorbs 70 J of heat. (a) How much heat does the gas exchange along BC? (b) When the gas makes the transition from C to A along CDA, 60 J of work are done on it from C to D. How much heat does it exchange along CDA?

Ideal gases A and B are stored in the left and right chambers of an insulated container, as shown below. The partition is removed and the gases mix. Is any work done in this process? If the temperatures of A and B are initially equal, what happens to their common temperature after they are mixed?

Consider the process for steam in a cylinder shown below. Suppose the change in the internal energy in this process is 30 kJ. Find the heat entering the system.

A monatomic ideal gas undergoes a quasi-static process that is described by the function $p(V)=$ $p_1+$ $3(V-V_1)$, where the pressure and volume are measured in SI units, the starting state is $\left(p_1,V_1\right)$, and the final state is $\left(p_2,V_2\right)$. Assume the system consists of n moles of the gas in a container that can exchange heat with the environment and whose volume can change freely. (a) Evaluate the work done by the gas during the change in the state. (b) Find the change in internal energy of the gas. (c) Find the heat input to the gas during the change. (d) What are initial and final temperatures?

A gas in a cylindrical closed container is adiabatically and quasi-statically expanded from a state A (3 MPa, 2 L) to a state B with volume of 6 L along the path ${pV}^{1.8}=$ $\text{constant}\text{.}$ (a) Plot the path in the pV plane. (b) Find the amount of work done by the gas and the change in the internal energy of the gas during the process.

Consider a transformation from point A to B in a two-step process. First, the pressure is lowered from 3 MPa at point A to a pressure of 1 MPa, while keeping the volume at 2 L by cooling the system. The state reached is labeled C. Then the system is heated at a constant pressure to reach a volume of 6 L in the state B. (a) Find the amount of work done on the ACB path. (b) Find the amount of heat exchanged by the system when it goes from A to B on the ACB path. (c) Compare the change in the internal energy when the AB process occurs isothermally with the AB change through the two-step process on the ACB path.

An ideal gas expands isothermally along AB and does 700 J of work (see below). (a) How much heat does the gas exchange along AB? (b) The gas then expands adiabatically along BC and does 400 J of work. When the gas returns to A along CA, it exhausts 100 J of heat to its surroundings. How much work is done on the gas along this path?

Two moles of helium gas are placed in a cylindrical container with a piston. The gas is at room temperature $25\text{°}\text{C}$ and under a pressure of $3.0\times {10}^5\text{Pa}\text{.}$ When the pressure from the outside is decreased while keeping the temperature the same as the room temperature, the volume of the gas doubles. (a) Find the work the external agent does on the gas in the process. (b) Find the heat exchanged by the gas and indicate whether the gas takes in or gives up heat. Assume ideal gas behavior.

The temperature of an ideal monatomic gas rises by 8.0 K. What is the change in the internal energy of 1 mol of the gas at constant volume?

If the gases of the preceding problem are initially at 300 K, what are their internal energies after they absorb the heat?

When 400 J of heat are slowly added to 10 mol of an ideal monatomic gas, its temperature rises by $10\text{°}\text{C}$. What is the work done on the gas?

A monatomic ideal gas undergoes a quasi-static adiabatic expansion in which its volume is doubled. How is the pressure of the gas changed?

Pressure and volume measurements of a dilute gas undergoing a quasi-static adiabatic expansion are shown below. Plot ln p vs. ln V and determine $\text{γ}$ for this gas from your graph.

P (atm)	V (L)
20.0	1.0
17.0	1.1
14.0	1.3
11.0	1.5
8.0	2.0
5.0	2.6
2.0	5.2
1.0	8.4

An ideal diatomic gas at 80 K is slowly compressed adiabatically and reversibly to half its volume. What is its final temperature?

Compare the charge in internal energy of an ideal gas for a quasi-static adiabatic expansion with that for a quasi-static isothermal expansion. What happens to the temperature of an ideal gas in an adiabatic expansion?

A dilute gas expands quasi-statically to three times its initial volume. Is the final gas pressure greater for an isothermal or an adiabatic expansion? Does your answer depend on whether the gas is monatomic, diatomic, or polyatomic?

On an adiabatic process of an ideal gas pressure, volume and temperature change such that $p{V}^{\gamma }$ is constant with $\gamma =5\text{/}3$ for monatomic gas such as helium and $\gamma =7\text{/}5$ for diatomic gas such as hydrogen at room temperature. Use numerical values to plot two isotherms of 1 mol of helium gas using ideal gas law and two adiabatic processes mediating between them. Use $T_1=$ $500\text{K},$ $V_1=$ $1\text{L},$ $\text{and}{T}_2=$ $300\text{K}$ for your plot.

Consider the process shown below. During steps AB and BC, 3600 J and 2400 J of heat, respectively, are added to the system. (a) Find the work done in each of the processes AB, BC, AD, and DC. (b) Find the internal energy change in processes AB and BC. (c) Find the internal energy difference between states C and A. (d) Find the total heat added in the ADC process. (e) From the information given, can you find the heat added in process AD? Why or why not?

A helium-filled rigid, sealed container has a gauge pressure of 0.200 atm and a volume of 10.0 L. How much greater is the internal energy of the helium in the container than it would be at zero gauge pressure? Assume the volume doesn't change and no gas leaks in or out.

A hand-driven tire pump has a piston with a 2.50-cm diameter and a maximum stroke of 30.0 cm. (a) How much work do you do in one stroke if the average gauge pressure is $2.4\times {10}^5{\text{N/m}}^2$ (about 35 psi)? (b) What average force do you exert on the piston, neglecting friction and gravitational force?

What is the net work output of a heat engine that follows path ABDA in the preceding problem with a straight line from B to D? Why is the work output less than for path ABCDA?

Four moles of a monatomic ideal gas in a cylinder at $27\text{°}\text{C}$ is expanded at constant pressure equal to 1 atm until its volume doubles. (a) What is the change in internal energy? (b) How much work was done by the gas in the process? (c) How much heat was transferred to the gas?

In an adiabatic process, oxygen gas in a container is compressed along a path that can be described by the following pressure in atm as a function of volume V, with $V_0=1L$: $p=$ $\left(3.0\text{atm}\right)(V\text{/}{V}_0{)}^{\mathrm{-1.2}}$. The initial and final volumes during the process were 2 L and 1.5 L, respectively. Find the amount of work done on the gas.

A cylinder containing three moles of nitrogen gas is heated at a constant pressure of 2 atm. The temperature of the gas changes from 300 K to 350 K as a result of the expansion. Find work done (a) on the gas, and (b) by the gas by using van der Waals equation of state instead of ideal gas law.

An insulated vessel contains 1.5 moles of argon at 2 atm. The gas initially occupies a volume of 5 L. As a result of the adiabatic expansion the pressure of the gas is reduced to 1 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the gas in the initial state. (c) Find the work done by the gas in the process. (d) Find the change in the internal energy of the gas in the process.

One mole of an ideal gas is initially in a chamber of volume $1.0\times {10}^{\mathrm{-2}}{\text{m}}^3$ and at a temperature of $27\text{°}\text{C}$. (a) How much heat is absorbed by the gas when it slowly expands isothermally to twice its initial volume? (b) Suppose the gas is slowly transformed to the same final state by first decreasing the pressure at constant volume and then expanding it isobarically. What is the heat transferred for this case? (c) Calculate the heat transferred when the gas is transformed quasi-statically to the same final state by expanding it isobarically, then decreasing its pressure at constant volume.

The insulated cylinder shown below is closed at both ends and contains an insulating piston that is free to move on frictionless bearings. The piston divides the chamber into two compartments containing gases A and B. Originally, each compartment has a volume of $5.0\times {10}^{\mathrm{-2}}{\text{m}}^3$ and contains a monatomic ideal gas at a temperature of $0\text{°}\text{C}$ and a pressure of 1.0 atm. (a) How many moles of gas are in each compartment? (b) Heat Q is slowly added to A so that it expands and B is compressed until the pressure of both gases is 3.0 atm. Use the fact that the compression of B is adiabatic to determine the final volume of both gases. (c) What are their final temperatures? (d) What is the value of Q?