To solve the problem step-by-step, let's break it down clearly. The goal is to analyze the given data and verify the given relationship according to the ideal gas law. The given quantities are:
1. Absolute temperature ([tex]\( T \)[/tex]) = 275 K
2. Pressure ([tex]\( P \)[/tex]) = 105 N/m²
3. Volume ([tex]\( V \)[/tex]) = 0.0225 m³
According to the ideal gas law, the volume [tex]\( V \)[/tex] of a given amount of gas varies directly with its absolute temperature [tex]\( T \)[/tex] and inversely with its pressure [tex]\( P \)[/tex]. The relationship can be expressed with the following formula:
[tex]\[ V \propto \frac{T}{P} \][/tex]
This means:
[tex]\[ V = k \frac{T}{P} \][/tex]
where [tex]\( k \)[/tex] is a proportionality constant. To find [tex]\( k \)[/tex], we'll use the provided values:
Given:
- [tex]\( V = 0.0225 \)[/tex] m³
- [tex]\( T = 275 \)[/tex] K
- [tex]\( P = 105 \)[/tex] N/m²
Using the formula, we can solve for [tex]\( k \)[/tex]:
[tex]\[ 0.0225 = k \frac{275}{105} \][/tex]
Solving the equation for [tex]\( k \)[/tex]:
1. Calculate the ratio [tex]\(\frac{275}{105}\)[/tex]:
[tex]\[ \frac{275}{105} = \frac{55}{21} \][/tex]
2. Substitute the ratio back into the equation:
[tex]\[ 0.0225 = k \cdot \frac{55}{21} \][/tex]
3. Isolate [tex]\( k \)[/tex]:
[tex]\[ k = 0.0225 \cdot \frac{21}{55} \][/tex]
4. Simplify the expression for [tex]\( k \)[/tex]:
[tex]\[ k = 0.0225 \cdot 0.3818 \approx 0.0086 \][/tex]
Therefore, [tex]\( k \approx 0.0086 \)[/tex]. This shows that:
[tex]\[ V = 0.0086 \cdot \frac{T}{P} \][/tex]
Given the proportionate relationship between volume, temperature, and pressure, the previously stated values for the specific gas (0.0225 m³, 275 K, and 105 N/m²) are verified by the ideal gas relationship. The calculated values confirm that if a gas has an absolute temperature of 275 K and pressure of 105 N/m², then the volume would correctly be [tex]\( 0.0225 \)[/tex] m³ according to the ideal gas law.
