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

검증된 단계별 안내

First, understand that the molar heat capacities at constant volume $ C_v $ and at constant pressure $ C_p $ for an ideal gas are related to the specific heat ratio $ \gamma $ (also denoted as $ g $ in this problem) by the formula $ \gamma = \frac{C_p}{C_v} $.

Recall the relationship between $ C_p $ and $ C_v $ for an ideal gas: $ C_p = C_v + R $, where $ R $ is the universal gas constant, approximately $ 8.314 \text{ J/mol·K} $.

Using the given $ \gamma = 1.127 $, set up the equation $ 1.127 = \frac{C_p}{C_v} $.

Substitute $ C_p = C_v + R $ into the equation $ 1.127 = \frac{C_v + R}{C_v} $ and solve for $ C_v $.

Once $ C_v $ is determined, use the equation $ C_p = C_v + R $ to find $ C_p $.

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주요 개념

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Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the behavior of ideal gases. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature. This law helps in understanding the relationship between these variables and is crucial for calculating properties like heat capacities.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a certain amount. Molar heat capacity, specifically, refers to the heat capacity per mole of a substance. It is important to distinguish between heat capacity at constant volume (Cv) and constant pressure (Cp), as they differ due to the work done by or on the system during expansion or compression.

Ratio of Heat Capacities (Gamma, γ)

The ratio of heat capacities, denoted as γ (gamma), is defined as Cp/Cv, where Cp is the molar heat capacity at constant pressure and Cv is the molar heat capacity at constant volume. For an ideal gas, this ratio is crucial for determining the individual heat capacities using the relation Cp = γCv. Understanding γ allows us to calculate these values, which are essential for thermodynamic processes.