Question

A certain mass of gas is expanded from (1L, 10 atm) to (4L, 5 atm) against a constant external pressure of 1 atm. If the initial temperature of the gas is 300 K and the heat capacity of the process is $ {50^o}C $ . Then the enthalpy change during the process is: (1L atm $ \simeq $ 100 J).
(A) $ \Delta H = 15{\text{ }}kJ $
(B) $ \Delta H = 15.7{\text{ }}kJ $
(C) $ \Delta H = 14.4{\text{ }}kJ $
(D) $ \Delta H = 14.7{\text{ }}kJ $

Answer 1

Hint :We can solve this question by using the Ideal gas equation and the first law of thermodynamics. Using the Ideal gas equation we can find the temperature at which the changes happen. Then we can use the expanded form of the first law and find out the change in internal energy of the given process. Then we can apply the equation for change in enthalpy

Complete Step By Step Answer:
First, we can use the Ideal gas equation and thus find the number of moles in state 1.
 $ \Rightarrow PV = nRT $
Here it is given that the pressure is 10 atm, volume is 1 litre and the temperature is 300 K.
 $ \Rightarrow 1 \times 10 = n(0.0821)(300) $
 $ \Rightarrow n = 0.4 $
Now for the second state we can use this value of n and find out T
 $ \Rightarrow PV = nRT $
 $ \Rightarrow 4 \times 5 = 0.4 \times (0.0821)({T_2}) $
 $ \Rightarrow {T_2} = 600K $
Now we can apply the first law of thermodynamics and the equation is as shown below:
 $ \Rightarrow \Delta U = q + W $
But we know that heat (q) can be written as the product of heat capacity and change in temperature.
W represents work and pressure times the change in volume and U is internal energy.
  $ \Rightarrow \Delta U = 50(\Delta T) + ( - {P_{ex}})(\Delta V) $
 $ \Rightarrow \Delta U = 50(600 - 300) - (1)(4 - 1) $
 $ \Rightarrow \Delta U = 15000 - 300 $
 $ \Rightarrow \Delta U\;\; = 14700\;J $
We know that the equation of change in enthalpy is as given below:
 $ \Rightarrow \Delta H = \Delta U + \Delta (PV) $
 $ \Rightarrow \Delta H = 14700 + (5 \times 4 - 10) $
 $ \Rightarrow \Delta H = 15.7\;kJ $
Thus we can say that the correct answer for this question will be option (B).

Note :
When we have to find the work done by the system then that will be negative and therefore is the negative value of the product of external pressure and change in volume. Since in question it is given that the gas is expanding we know that it has to take heat from the system to do that work. Remember that heat is taken from a system is always negative whereas given to the system is positive.