The volume of a gas is halved during an adiabatic compression that increases the pressure by a factor of 2.5. What is the specific heat ratio γ?

Verified step by step guidance

1. Step 1: Understand the problem. The process described is an adiabatic compression, meaning no heat is exchanged with the surroundings. The relationship between pressure, volume, and the specific heat ratio γ in an adiabatic process is given by the equation: $P_2=P_1\cdot {\left(}^{\frac{V_1}{V_2}}$, where P is pressure, V is volume, and γ is the specific heat ratio.
2. Step 2: Substitute the given values into the equation. The pressure increases by a factor of 2.5, so $P_2=2.5\cdot P_1$. The volume is halved, so $V_2=\frac{V_1}{2}$. Replace these values in the equation: $2.5={\left(}^2$.
3. Step 3: Solve for γ. To isolate γ, take the natural logarithm (ln) of both sides of the equation: $\ln \left(2.5\right)=\gamma \cdot \ln \left(2\right)$. Rearrange to find γ: $\gamma =\frac{\ln }{\left(}$.
4. Step 4: Use logarithmic properties to simplify the calculation. Compute the natural logarithms of 2.5 and 2 separately, and then divide them to find γ. This step involves numerical computation, which is not performed here.
5. Step 5: Interpret the result. The specific heat ratio γ is a dimensionless quantity that characterizes the thermodynamic properties of the gas. It is typically greater than 1 and depends on the type of gas (e.g., monoatomic, diatomic, or polyatomic).

Key Concepts

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. In such processes, any change in the internal energy of the system is due solely to work done on or by the system. This concept is crucial for understanding how gases behave under compression or expansion without heat transfer.

Specific Heat Ratio (γ)

The specific heat ratio, denoted as γ (gamma), is the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv). It is a dimensionless quantity that characterizes the thermodynamic properties of gases. For ideal gases, γ is important in determining how pressure, volume, and temperature change during adiabatic processes.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and 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 provides a basis for understanding the behavior of gases during various thermodynamic processes, including adiabatic compression.