For nitrogen gas, cv=740 J / kg ·K. Assuming it to behave like an ideal gas, find its specific

step 1 we have to find the value of specific heat at constant pressure for nitrogen gas solution given

For nitrogen gas, cv=740 J / kg ·K. Assuming it to behave...

For nitrogen gas, $c_{v}=740 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Assuming it to behave like an ideal gas, find its specific heat at constant pressure. (The molecular mass of nitrogen gas is $28.0 \mathrm{~kg} / \mathrm{kmol} .$ ) Method 1 $$c_{p}=c_{v}+\frac{R}{M}=\frac{740 \mathrm{~J}}{\mathrm{~kg} \cdot \mathrm{K}}+\frac{8314 \mathrm{~J} / \mathrm{kmol} \cdot \mathrm{K}}{28.0 \mathrm{~kg} / \mathrm{kmol}}=1.04 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$$ Method 2 Since $\mathrm{N}_{2}$ is a diatomic gas, and since $\gamma=c_{p} / c_{v}$ for such a gas, $$ c_{p}=1.40 c_{v}=1.40(740 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K})=1.04 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K} $$

For nitrogen gas, $c_{v}=740 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Assuming it to behave like an ideal gas, find its specific heat at constant pressure. (The molecular mass of nitrogen gas is $28.0 \mathrm{~kg} / \mathrm{kmol} .$ )
Method 1
$$c_{p}=c_{v}+\frac{R}{M}=\frac{740 \mathrm{~J}}{\mathrm{~kg} \cdot \mathrm{K}}+\frac{8314 \mathrm{~J} / \mathrm{kmol} \cdot \mathrm{K}}{28.0 \mathrm{~kg} / \mathrm{kmol}}=1.04 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$$
Method 2
Since $\mathrm{N}_{2}$ is a diatomic gas, and since $\gamma=c_{p} / c_{v}$ for such a gas,
$$
c_{p}=1.40 c_{v}=1.40(740 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K})=1.04 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}
$$

Key Concepts

Molecular Mass and Unit Conversion

Molecular mass is used to convert between molar quantities and mass-specific quantities. In calculations involving the universal gas constant, converting R from a molar basis to a per unit mass basis ensures that units are consistent and the physical interpretation of thermodynamic properties remains valid.

Universal Gas Constant (R)

The universal gas constant R is a fundamental constant that appears in the ideal gas law, providing a bridge between the macroscopic and microscopic descriptions of gas behavior. It allows conversion between properties measured per mole and per unit mass, which is essential when using specific heats expressed in energy per unit mass.

Specific Heat Ratio (?)

The specific heat ratio, defined as ? = cp/cv, is a dimensionless number that characterizes the thermodynamic behavior of gases, particularly in processes like adiabatic expansion and compression. For diatomic gases, this ratio is typically around 1.4, reflecting the degrees of freedom associated with rotational modes in addition to translational energy.

Specific Heats (cv and cp)

Specific heats represent the amount of energy required to raise the temperature of a unit mass of a substance by one degree under specific conditions. cv is defined under constant volume and cp under constant pressure. In thermodynamic processes, understanding the difference between these two is crucial because the work done during expansion at constant pressure means cp is always larger than cv for an ideal gas.

Relationship Between cp and cv in an Ideal Gas

For an ideal gas, cp and cv are related by the equation cp = cv + (R/M), where R is the universal gas constant and M is the molecular mass. This relationship stems from the first law of thermodynamics and the additional work done during constant pressure processes, effectively linking microscopic energy contributions to macroscopic properties.

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