You compress a gas in an insulated cylinder-no heat flows...

You compress a gas in an insulated cylinder-no heat flows into or out of the gas. The gas pressure is fairly low, so treating the gas as ideal is a good approximation. When you measure the pressure as a function of the volume of the gas, you obtain these results: $$ \begin{array}{l|lllll} \boldsymbol{V}(\mathbf{L}) & 2.50 & 2.02 & 1.48 & 1.01 & 0.50 \\ \hline p(\mathrm{~atm}) & 0.101 & 0.139 & 0.202 & 0.361 & 0.952 \end{array} $$ (a) Graph $\log (p)$ versus $\log (V)$, with $p$ in $\mathrm{Pa}$ and $V$ in $\mathrm{m}^{3}$. Explain why the data points fall close to a straight line. (b) Use your graph to calculate $\gamma$ for the gas. Is the gas monatomic, diatomic, or polyatomic? (c) When $p=0.101 \mathrm{~atm}$ and $V=2.50 \mathrm{~L},$ the temperature is $22.0^{\circ} \mathrm{C}$. Apply the ideal-gas equation and calculate the temperature for each of the other pairs of $p$ and $V$ values. In this compression, does the temperature of the gas increase, decrease, or stay constant?

You compress a gas in an insulated cylinder-no heat flows into or out of the gas. The gas pressure is fairly low, so treating the gas as ideal is a good approximation. When you measure the pressure as a function of the volume of the gas, you obtain these results:
$$
\begin{array}{l|lllll}
\boldsymbol{V}(\mathbf{L}) & 2.50 & 2.02 & 1.48 & 1.01 & 0.50 \\
\hline p(\mathrm{~atm}) & 0.101 & 0.139 & 0.202 & 0.361 & 0.952
\end{array}
$$
(a) Graph $\log (p)$ versus $\log (V)$, with $p$ in $\mathrm{Pa}$ and $V$ in $\mathrm{m}^{3}$. Explain why the data points fall close to a straight line. (b) Use your graph to calculate $\gamma$ for the gas. Is the gas monatomic, diatomic, or polyatomic? (c) When $p=0.101 \mathrm{~atm}$ and $V=2.50 \mathrm{~L},$ the temperature is $22.0^{\circ} \mathrm{C}$. Apply the ideal-gas equation and calculate the temperature for each of the other pairs of $p$ and $V$ values. In this compression, does the temperature of the gas increase, decrease, or stay constant?

Key Concepts

Adiabatic Process

An adiabatic process is one in which there is no heat exchange between the system and its surroundings. Instead, any change in the internal energy of the gas is due solely to work done on or by the system. This process leads to a unique relationship between pressure, volume, and temperature that differs from those in isothermal or other processes where heat is exchanged.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. This law assumes that the gas particles do not interact except via elastic collisions and that the volume of the particles is negligible compared to the volume of the container. It serves as a foundational model for understanding gas behavior under a variety of conditions.

Specific Heat Ratio (?)

The specific heat ratio, denoted as ?, is the ratio of the heat capacity at constant pressure (Cp) to the heat capacity at constant volume (Cv). In adiabatic processes for an ideal gas, ? appears in the relation pV^? = constant. Its value helps classify the type of gas (monatomic, diatomic, or polyatomic) according to the degrees of freedom available to the gas molecules.

Power Law Relationship

A power law relationship is a functional relationship between two quantities where one quantity is proportional to a power of the other. In thermodynamics, the relationship pV^? = constant is a power law, with the exponent being the specific heat ratio ?. This power law can be linearized by taking the logarithm of both sides, converting the relationship into a straight line equation on a log-log plot.

Log-Log Plot Analysis

A log-log plot involves graphing the logarithm of one variable against the logarithm of another. In systems governed by power laws, such as the adiabatic relation pV^? = constant, the log-log plot transforms the power-law equation into a linear relationship, making it easier to determine the exponent from the slope of the resulting line. This method is widely used for identifying and analyzing power law behaviors in various fields, including thermodynamics.

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