One mole of an ideal monatomic gas undergoes a process described by the equation $PV^{3}=$ constant. The heat capacity of the gas during this process is -
- $\frac{3}{2}R$
- $\frac{5}{2}R$
- $2 \, R$
- $R$
The Correct Option is D
Approach Solution - 1
Specific Heat in Polytropic Process
Given the equation for the polytropic process:
\(PV^{n} = K \, \text{where} \, n = 3\)
The equation for specific heat in the polytropic process is:
\(C = C_V + \frac{R}{1 - n}\)
Substituting the values:
\(C = \frac{3}{2} R + \frac{R}{1 - 3}\)
Solving the equation:
\(C = \frac{3}{2} R - \frac{R}{2} = R\)
Thus, the specific heat for this process is equal to \(R\).
Approach Solution -2
Heat Capacity of an Ideal Monatomic Gas
Given the process described by the equation:
\(P V^3 = \text{constant}\)
This represents a polytropic process where the polytropic index \(n = 3\).
To calculate the heat capacity \(C\) during this process, we use the formula:
\(C = C_V + \frac{R}{1 - n}\)
For a monatomic ideal gas, the specific heat at constant volume \(C_V\) is:
\(C_V = \frac{3}{2} R\)
Now, substituting \(n = 3\) into the equation for heat capacity:
\(C = \frac{3}{2} R + \frac{R}{1 - 3} = \frac{3}{2} R + \frac{R}{-2}\)
Simplifying this gives:
\(C = \frac{3}{2} R - \frac{R}{2} = R\)
Thus, the heat capacity \(C\) during this process is equal to R.
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Concepts Used:
Enthalpy change
Enthalpy Change refers to the difference between the heat content of the initial and final state of the reaction. Change in enthalpy can prove to be of great importance to find whether the reaction is exothermic or endothermic.
Formula for change in enthalpy is:-
dH = dU + d(PV)
The above equation can be written in the terms of initial and final states of the system which is defined below:
UF – UI = qP –p(VF – VI)
Or qP = (UF + pVF) – (UI + pVI)
Enthalpy (H) can be written as H= U + PV. Putting the value in the above equation, we obtained:
qP = HF – HI = ∆H
Hence, change in enthalpy ∆H = qP, referred to as the heat consumed at a constant pressure by the system. At constant pressure, we can also write,
∆H = ∆U + p∆V
Standard Enthalpy of Reaction
To specify the standard enthalpy of any reaction, it is calculated when all the components participating in the reaction i.e., the reactants and the products are in their standard form. Therefore the standard enthalpy of reaction is the enthalpy change that occurs in a system when a matter is transformed by a chemical reaction under standard conditions.