Propane gas (C3H8) behaves like an ideal gas with g = 1.127. Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure

Verified step by step guidance

1. Identify the specific heat ratio (g) for propane, which is given as 1.127. This ratio is also known as the adiabatic index and is denoted by \\( ext{g}\\).
2. Recall the relationship between the molar heat capacity at constant volume (\\(C_V\\)) and the molar heat capacity at constant pressure (\\(C_P\\)) for an ideal gas, which is given by the equation \\( ext{g} = \frac{C_P}{C_V}\\).
3. Use the universal gas constant \\( ext{R}\\), which is approximately 8.314 J/(mol·K), to relate \\( ext{C}_P\\) and \\( ext{C}_V\\). The relationship is \\( ext{C}_P = ext{C}_V + ext{R}\\).
4. Substitute the expression for \\( ext{C}_P\\) from step 3 into the equation from step 2 to solve for \\( ext{C}_V\\). Rearrange the equation to get \\( ext{C}_V = \frac{ ext{R}}{ ext{g} - 1}\\).
5. Substitute the value of \\( ext{C}_V\\) back into the equation \\( ext{C}_P = ext{C}_V + ext{R}\\) to find \\( ext{C}_P\\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law describes the relationship between pressure, volume, temperature, and the number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law is fundamental for understanding the behavior of gases under various conditions and is essential for calculating properties like heat capacities.

Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant volume (C_v) or constant pressure (C_p). C_v is typically lower than C_p because at constant pressure, work is done by the system as it expands. Understanding these concepts is crucial for thermodynamic calculations involving gases.

Specific Heat Ratio (γ)

The specific heat ratio, denoted as γ (gamma), is the ratio of molar heat capacities at constant pressure and volume (C_p/C_v). For ideal gases, this ratio is a key parameter that influences the behavior of the gas during processes such as adiabatic expansion or compression. Knowing γ helps in deriving relationships between heat capacities and is vital for solving problems related to thermodynamics.