Adiabatic Compression: Temperature Change Calculation

This solution explains the calculation for the temperature change of a gas during an adiabatic compression process.

Problem Analysis

The problem involves a gas contained in a thermally non-conducting container, implying an adiabatic process. Key parameters are:

• Initial Volume ($V_1$): $800 \ cm^3$
• Initial Temperature ($T_1$): $27^{\circ}C$. Convert to Kelvin: $T_1 = 27 + 273 = 300 \ K$.
• Final Volume ($V_2$): $200 \ cm^3$
• Adiabatic Index ($\gamma$): $1.5$
• Goal: Calculate the change in temperature ($\Delta T$).

Adiabatic Process Equation

For an adiabatic process involving an ideal gas, the relationship between temperature and volume is constant:

$ T V^{\gamma-1} = \text{constant} $

Therefore, for the initial and final states:

$ T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} $

Calculating Final Temperature ($T_2$)

Rearrange the equation to solve for the final temperature ($T_2$):

$ T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma-1} $

Substitute the known values:

$ T_2 = 300 \ K \times \left( \frac{800 \ cm^3}{200 \ cm^3} \right)^{1.5 - 1} $

Simplify the volume ratio and the exponent:

$ \frac{V_1}{V_2} = 4 $

$ \gamma - 1 = 0.5 $

Now, calculate $T_2$:

$ T_2 = 300 \ K \times (4)^{0.5} $

$ T_2 = 300 \ K \times \sqrt{4} $

$ T_2 = 300 \ K \times 2 $

$ T_2 = 600 \ K $

Calculating Temperature Change ($\Delta T$)

The required value is the change in temperature ($\Delta T$), calculated as the final temperature minus the initial temperature:

$ \Delta T = T_2 - T_1 $

$ \Delta T = 600 \ K - 300 \ K $

$ \Delta T = 300 \ K $

Result

The change in temperature for the gas during adiabatic compression is $300 \ K$.