Question

Pressure over 1000 ml of a liquid is gradually increasing from 1 bar to 1001 bar under adiabatic conditions. If the final volume of the liquid is 990 ml. Calculate $\Delta U$ and $\Delta H$ of the process, assuming linear variation of volume with pressure.

Answer 1

Hint: The formulas that are used to solve this question are then the relation between the change in enthalpy and change in internal energy as:
$\Delta H=\Delta U+P\Delta V$
And there is property which will be calculated, i.e., work done. Its formula is:
$w=\dfrac{\text{Total pressure x change in volume}}{2}$

Complete answer:
The given system is an adiabatic system, so the change in heat of the system will be constant. Therefore, according to the first law of thermodynamics:
$\Delta U=w-q$
q will be constant, so the work done on the system will be equal to the change in the internal energy.
$\Delta U=w$
We can find the word done for the adiabatic system as:
$w=\dfrac{\text{Total pressure x change in volume}}{2}$
Initial pressure ${{P}_{1}}$ is 1 bar
Final pressure ${{P}_{2}}$ is 1001 bar
Initial volume ${{V}_{1}}$ is 1000 ml
Final volume ${{V}_{2}}$ is 990 ml
So, the work done can be calculated as:
$w=\dfrac{1}{2}\text{ x (1001+1) x (1000-990) = 5010 bar mL}$
The work done of the system is 5010 bar mL, dividing this by 1000, we can get the work done in kilojoules.
Work done = 5.01 kJ.
So, the change in internal energy $\Delta U$ is 5.01 kJ.
The change in the enthalpy of the system can be calculated as:
$\Delta H=\Delta U+P\Delta V$
This can be written as:
$\Delta H=\Delta U+({{P}_{2}}{{V}_{2}}-{{P}_{1}}{{V}_{1}})$
We can directly take the volume in liters. Putting the values, we get:
$\Delta H=5.01+(1001\text{ x 0}\text{.99}-\text{ 1 x 1})\text{ x 100}$
$\Delta H=99500J$
We can convert this into kilojoules as:
$\Delta H=99.5\text{ kJ}$

Note:
Whenever the given process in the question is adiabatic then the heat or change in heat of the system will be constant and for calculating the first law of thermodynamics, the word done and internal energy will be equal.