Sure, let's break down the steps to find the factor by which the volume increases when an ideal gas is allowed to expand from 6.40 liters to 60.8 liters at constant temperature:
1. Identify Initial and Final Volumes:
The initial volume of the gas is 6.40 liters, and the final volume of the gas is 60.8 liters. We denote these volumes as [tex]\( V_{\text{initial}} \)[/tex] and [tex]\( V_{\text{final}} \)[/tex] respectively.
2. Calculate the Volume Increase Factor:
The factor by which the volume increases is given by the ratio of the final volume to the initial volume. Mathematically, this is expressed as:
[tex]\[
\text{Volume Increase Factor} = \frac{V_{\text{final}}}{V_{\text{initial}}}
\][/tex]
3. Substitute the Given Values:
- [tex]\( V_{\text{initial}} = 6.40 \, \text{L} \)[/tex]
- [tex]\( V_{\text{final}} = 60.8 \, \text{L} \)[/tex]
Plugging these values into the formula:
[tex]\[
\text{Volume Increase Factor} = \frac{60.8 \, \text{L}}{6.40 \, \text{L}}
\][/tex]
4. Simplify the Expression:
Simplifying the ratio:
[tex]\[
\text{Volume Increase Factor} = 9.499999999999998
\][/tex]
Therefore, the factor by which the volume increases is approximately 9.5.
5. Effect on Pressure:
According to Boyle's Law, for an ideal gas at constant temperature, the pressure and volume of the gas are inversely proportional. This means if the volume increases by a factor of 9.5, the pressure will decrease by the same factor.
This is because:
[tex]\[
P \propto \frac{1}{V}
\][/tex]
Therefore, given that the volume has increased by a factor of 9.5, the pressure will decrease by that same factor.
In summary, the volume of the gas increases by a factor of approximately 9.5, and as a result, the pressure decreases by that same factor.
