The ratio of specific heat (Cp/Cv) of Nitrogen N2 is 1.40. Nitrogens gas constant is R = 296.8 [J/kgK]. Determine the constant volume specific heat Cv and constant pressure specific heat Cp of Nitrogen.
The correct answer and explanation is :
To determine the specific heats at constant volume ((C_v)) and constant pressure ((C_p)) for nitrogen ((N_2)), we use the following thermodynamic relations:
- Ratio of Specific Heats (Given):
[
\gamma = \frac{C_p}{C_v} = 1.40
] - Gas Constant (Given):
[
R = 296.8 \text{ J/kgK}
] - Relationship Between Specific Heats and Gas Constant:
[
C_p – C_v = R
]
Step 1: Express ( C_p ) in Terms of ( C_v )
From the definition of (\gamma):
[
C_p = \gamma C_v
]
Substituting (\gamma = 1.40):
[
C_p = 1.40 C_v
]
Step 2: Solve for ( C_v )
Using the equation (C_p – C_v = R):
[
(1.40 C_v) – C_v = 296.8
]
[
0.40 C_v = 296.8
]
[
C_v = \frac{296.8}{0.40} = 742 \text{ J/kgK}
]
Step 3: Solve for ( C_p )
[
C_p = 1.40 \times 742 = 1038.8 \text{ J/kgK}
]
Final Answer:
- Constant Volume Specific Heat ((C_v)) = 742 J/kgK
- Constant Pressure Specific Heat ((C_p)) = 1038.8 J/kgK
Explanation:
The specific heats (C_p) and (C_v) are fundamental properties of gases. (C_v) represents the amount of heat required to raise the temperature of 1 kg of nitrogen by 1K at constant volume, whereas (C_p) represents the same process at constant pressure. The relationship (C_p – C_v = R) arises from the first law of thermodynamics applied to an ideal gas. Since nitrogen behaves as a diatomic ideal gas, its (\gamma) value is typically around 1.40. By using this ratio and the given gas constant, we solve for (C_v) and then determine (C_p).