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The problem describes an experiment where the relationship between the pressure \( P \) and volume \( V \) of a gas is given, assuming adiabatic conditions, by:

\[
pV^\gamma = C
\]

where:
- \( p \) is the pressure,
- \( V \) is the volume,
- \( \gamma \) (gamma) is the adiabatic index,
- \( C \) is a constant.

The table of pressure \( P \) (in units of \( 10^5 \, \text{N/m}^2 \)) and volume \( V \) (in cubic meters) is provided as:

\[
\begin{array}{c|cccccc}
P (10^5 \, \text{N/m}^2) & 1.5 & 1.4 & 1.3 & 1.2 & 1.1 & 1.0 \\
V (\text{m}^3) & 0.95 & 1.00 & 1.05 & 1.10 & 1.16 & 1.24 \\
\end{array}
\]

You are asked to find reasonable values of \( \gamma \) and \( C \) that fit the given data and then predict the volume \( V \) when the pressure \( p \) is equal to atmospheric pressure \( 1.013 \times 10^5 \, \text{N/m}^2 \).

### Approach:
1. **Fit the data**: Use the equation \( pV^\gamma = C \) and apply logarithms to linearize the relationship. This will allow solving for \( \gamma \) using the data points.
   
2. **Determine \( \gamma \)**: By finding the slope of the linearized equation (logarithmic form), you can estimate the value of \( \gamma \).

3. **Calculate \( C \)**: Once \( \gamma \) is known, substitute any pair of \( p \) and \( V \) to find \( C \).

4. **Prediction at atmospheric pressure**: Use the values of \( \gamma \) and \( C \) to predict \( V \) for the given pressure \( p = 1.013 \times 10^5 \, \text{N/m}^2 \).

I'll compute these values next.The results of the analysis are as follows:

- **Fitted value of \( \gamma \)**: \( 1.546 \)
- **Fitted value of \( C \)**: \( 1.393 \times 10^5 \, \text{N/m}^{2} \cdot \text{m}^{\gamma} \)
- **Predicted volume at atmospheric pressure \( p = 1.013 \times 10^5 \, \text{N/m}^2 \)**: \( V \approx 1.229 \, \text{m}^3 \)

This implies that at atmospheric pressure, the predicted volume of the gas is approximately \( 1.229 \, \text{m}^3 \).

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions to consider:
1. How is the adiabatic index \( \gamma \) related to the type of gas?
2. What is the difference between adiabatic and isothermal processes?
3. How would the results differ if the process were isothermal instead of adiabatic?
4. How could you experimentally determine the value of \( \gamma \) for other gases?
5. How does the value of \( C \) depend on the initial conditions of the gas?

**Tip:** In adiabatic processes, no heat is exchanged with the environment, meaning that the internal energy change of the gas is directly related to work done on or by the gas.

An experiment measuring the change in volume of a gas as the pressure is decreased gave the following measurements: Pressure (P in 10^5 N/m^2) and Volume (V in m^3). If the gas is assumed to be ideal and the expansion is adiabatic, the relationship between pressure and volume should be pV^γ = C. Find reasonable values of γ and C to fit the data and predict the volume at atmospheric pressure (p = 1.013 x 10^5 N/m^2).

Learn how to fit data for an adiabatic expansion experiment, where pressure and volume of a gas are measured. Calculate γ and C using the adiabatic equation pV^γ = C and predict the volume at atmospheric pressure p = 1.013 x 10^5 N/m^2.

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