Question

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100K and the universal gas constant R = 8.0 \[Jmo{{l}^{-1}}{{K}^{-1}}\], the decrease in its internal energy, in Joule, is

Answer 1

Hint: We are given with ideal gas, so, we can use the ideal gas equation here. The moles are given to be one mole, the volume changes and becomes eight-time of its initial value. The initial temperature is given. We need to find a decrease in internal energy given this to be an adiabatic expansion.

Complete step by step answer:
Moles, n= 1
The initial volume, \[{{V}_{i}}\]= V
The final volume, \[{{V}_{f}}\]= 8V
Temperature, T= 100 K
Gas constant, R = 8.0 \[Jmo{{l}^{-1}}{{K}^{-1}}\]
This is an adiabatic expansion and, in such expansion, we can use the equation for it.
Using, \[{{T}_{i}}V_{i}^{\gamma -1}={{T}_{2}}V_{2}^{\gamma -1}\]
We can find out the final temperature of the gas. Since this is a monatomic gas, so, the value of \[\gamma \] is \[\dfrac{5}{3}\]
Putting the values,
\[\begin{align}
  & 100\times {{V}^{\dfrac{2}{3}}}={{T}_{2}}{{(8V)}^{\dfrac{2}{3}}} \\
 & {{T}_{2}}=\dfrac{100}{4}=25K \\
\end{align}\]
So the final temperature is 25 K. the change in temperature is 100-25= 75 K
Now change in internal energy can be found out using the formula \[\Delta U=n{{C}_{v}}\Delta T\]& for a monatomic ideal gas the value of \[{{C}_{v}}=\dfrac{3R}{2}\].
\[\Delta U=1\times \dfrac{3\times 8}{2}\times 75=900J\]

So, the change in internal energy comes out to be 900 J.

Note:
We have used ideal gas law here because it was mentioned in the question that the gas is ideal. Otherwise, we would have to use real gas laws. Also, we had taken the value of the gas constant to be 8, if otherwise not mentioned we would have to use the original value of 8.314.