Question

Question: (II) In an engine, an almost ideal gas is compressed adiabatically to half its volume. In doing so, 2630 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall?

Short answer

(a)The heat that flows into or out of the gas is zero.

(b) The change in the internal energy of the gas is \(2630\;{\rm{J}}\).

(c) The temperature rises during the adiabatic compression process.

Step-by-step solution

First law of thermodynamics

According to the first law of thermodynamics, the change in the closed system’s internal energy can be expressed as:

\(\Delta U = Q - W\)

Here,\(Q\)is the heat added and\(W\)is the work done.

Given information

Given data:

The work done on the gas is \(W = - 2630\;{\rm{J}}\).

Evaluation of quantity of heat that flows into or out of the gas

(a)

There is no heat exchange happens between the surroundings and system in the adiabatic compression process. Hence, the heat that flows into or out of the gas is zero. That is \(Q = 0\).

Evaluation of internal energy of the gas

(b)

The change in the internal energy of the gas can be calculated as:

\(\begin{aligned}{c}\Delta U &= Q - W\\\Delta U &= 0 - \left( { - 2630\;{\rm{J}}} \right)\\\Delta U &= 2630\;{\rm{J}}\end{aligned}\)

Thus, the change in the internal energy of the gas is \(2630\;{\rm{J}}\).

Temperature variation in adiabatic compression process

(c)

As the gas is compressed adiabatically, there will be an increment in the temperature of the gas. Moreover, from part (b), it is clear that the change in internal energy is positive, which indicates a rise in internal energy. Temperature varies directly with internal energy. Therefore, as internal energy rises, the temperature also rises.

Hence, the temperature rises during the adiabatic compression process.