Chapter 19, Problem 1Q

For four situations for an ideal gas, the table gives the energy transferred to or front the gas as heat Q and either the work W done by the gas or the work W_on done on the gas, all in joules. Rank the four situations in terms of the temperature change of the gas, most positive first.

To determine

To rank:

The four situations in terms of temperature change of the gas, most positive first.

Answer to Problem 1Q

Solution:

The ranking of four situations in terms of temperature change of the gas is

$d>a=b>c$

Explanation of Solution

1) Concept:

We can find the change in the internal energy using the first law of thermodynamics. Then using the relation between change in internal energy and change in temperature, we can rank the four situations in terms of temperature change of the gas.

2) Formulae:

$∆E=Q-W$

$∆E=n{C}_V∆T$

3) Given:

The table containing four situations for an ideal gas

4) Calculations:

The change in the internal energy is given by

$∆E=n{C}_V∆T$

So, change in the internal energy of an ideal gas depends solely on the change in temperature. According to the first law of thermodynamics,

$∆E=Q-W$

where, heat energy Q is taken positive if it is added to the system and work done W is taken as positive if it is done by the gas.

a) In situation a,

$∆E=-50-(-50)$

$∆E=0$

Hence,

$∆T=0$

b) In situation b,

$∆E=35-(+35)$

$∆E=0$

Hence,

$∆T=0$

c) In situation c,

$∆E=-15-(+40)$

$∆E=25$

Hence,

$∆T<0$

d) In situation d,

$∆E=20-(-40)$

$∆E=60$

Hence,

$∆T>0$

Therefore, the ranking of four situations in terms of temperature change of the gas is,

$d>a=b>c$

Conclusion:

The change in the internal energy of an ideal gas solely depends on the change in temperature.