Question

5 moles of an ideal gas expand isothermally and reversibly from a pressure of 10atm to 2atm at 300 K. What is the mass which can be lifted through a height of 1 metre in this expansion?

Answer 1

Hint:We are given with the different factors, and with the help of that we can calculate the work done in isothermal reversible expansion. Then, apply the formula related to work done in terms of height. The mass can be determined.

Complete step by step solution:
> First, we will discuss the work done in isothermal, and reversible expansion.
- When we will consider the it for an ideal gas, the temperature is constant, i.e. $\Delta$T = 0, and the internal energy is also constant, i.e. $\Delta$U = 0.
- It can be written as $\Delta$U = q + W, then q = -W.
So, the work done in isothermal reversible expansion in terms of volume is represented as
W = - 2.303nRT log$\dfrac{V_2}{V_1}$
We know, P$_1$V$_1$=P$_2$V$_2$, as PV = nRT
So, W = - 2.303nRT log$\dfrac{P_1}{P_2}$
> According to the question, initial pressure (P$_1$)10 atm, and final pressure (P$_2$) 2 atm, temperature is 300 K. If we substitute the values, then
W = -2.303 $\times$ 5 $\times$ 8.314 $\times$ 300 log $\dfrac{10}{2}$ = -20.075$\times$ 10$^{3}$ J (R is gas constant with the value 8.314)
- Thus, work done in reversible isothermal expansion is -20.075 $\times$ 10$^{3}$ J.
- Now, we have to find the mass, consider the mass to be M thru which it can be lifted to a height of 1 m.
So, Work done = Mgh = M $\times$ 9.8 $\times$ 1
Thus, 20.075$\times$ 10$^{3}$ J = M $\times$ 9.8
Therefore, we have M = 2048.469 kg
In the last, we can conclude that mass that can be lifted thru a height of 1 m in this expansion is 2048.469kg.

Note: Don’t get confused while finding the mass lifted thru a height of 1 m. The unit of mass can be written in kg or g, as the unit is not mentioned in the question. Thus, we can calculate in both the units. While calculating the mass write down the work done equation in terms of pressure, and the work done calculate will help to determine the mass. The negative sign is not considered as mass cannot be negative.