Certainly! Let's work through this problem step-by-step using the Ideal Gas Law. The equation relating volume [tex]\( V \)[/tex], temperature [tex]\( T \)[/tex], and pressure [tex]\( P \)[/tex] for an ideal gas is given by:
[tex]\[ V = k \frac{T}{P} \][/tex]
where [tex]\( k \)[/tex] is a proportionality constant which needs to be determined.
### Given Data:
- Volume [tex]\( V = 0.225 \, \text{m}^2 \)[/tex]
- Temperature [tex]\( T = 275 \, \text{K} \)[/tex]
- Pressure [tex]\( P = 105 \, \text{N/m}^2 \)[/tex]
### Step-by-Step Solution:
1. Substitute the given values into the Ideal Gas Law:
We have [tex]\( V = k \frac{T}{P} \)[/tex].
[tex]\[ 0.225 = k \frac{275}{105} \][/tex]
2. Solve for the constant [tex]\( k \)[/tex]:
Multiply both sides of the equation by [tex]\( \frac{105}{275} \)[/tex]:
[tex]\[ k = 0.225 \times \frac{105}{275} \][/tex]
3. Perform the multiplication:
Calculate the product:
[tex]\[ k \approx 0.225 \times 0.3818 \][/tex]
### Result:
[tex]\[ k \approx 0.0859090909 \][/tex]
So, the proportionality constant [tex]\( k \)[/tex] in the equation [tex]\( V = k \frac{T}{P} \)[/tex] is approximately [tex]\( 0.0859090909 \)[/tex].
### Conclusion:
Given the specified temperature and pressure, the formula for the volume [tex]\( V \)[/tex] in terms of temperature [tex]\( T \)[/tex] and pressure [tex]\( P \)[/tex] is:
[tex]\[ V = 0.0859090909 \frac{T}{P} \][/tex]
This equation describes how the volume changes with varying temperature and pressure for the given ideal gas, based on the constants provided.
