Certainly! Let's walk through the detailed step-by-step solution for this problem using the Ideal Gas Law and the principles of gas behavior:
### Problem Statement
We have certain properties of an ideal gas given:
- Initial pressure [tex]\( P_1 = 105 \, \text{N/m}^2 \)[/tex]
- Initial volume [tex]\( V_1 = 0.0225 \, \text{m}^3 \)[/tex]
- Initial temperature [tex]\( T_1 = 275 \, \text{K} \)[/tex]
### Question
We need to determine how these properties relate to each other using the Ideal Gas Law.
### Ideal Gas Law
The Ideal Gas Law states that:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure of the gas,
- [tex]\( V \)[/tex] is the volume of the gas,
- [tex]\( n \)[/tex] is the number of moles of the gas,
- [tex]\( R \)[/tex] is the universal gas constant,
- [tex]\( T \)[/tex] is the temperature of the gas in Kelvin.
For a given amount of gas, the ratio [tex]\( \frac{PV}{T} \)[/tex] is a constant. This is derived from the fact that [tex]\( n \)[/tex] and [tex]\( R \)[/tex] are constants for a given sample.
### Calculation of the Constant
Given the initial conditions [tex]\( P_1 \)[/tex], [tex]\( V_1 \)[/tex], and [tex]\( T_1 \)[/tex], we can calculate the value of this proportional constant [tex]\( k \)[/tex] using:
[tex]\[ k = \frac{P_1 V_1}{T_1} \][/tex]
### Substituting the Given Values
Let's substitute the given values into the equation:
- [tex]\( P_1 = 105 \, \text{N/m}^2 \)[/tex]
- [tex]\( V_1 = 0.0225 \, \text{m}^3 \)[/tex]
- [tex]\( T_1 = 275 \, \text{K} \)[/tex]
So,
[tex]\[ k = \frac{105 \times 0.0225}{275} \][/tex]
### Calculation
Performing the multiplication and division, we get:
[tex]\[ k = \frac{2.3625}{275} \][/tex]
[tex]\[ k = 0.00859090909090909 \][/tex]
### Final Result
Thus, the relationship or the constant for this case, confirmed by calculations of the Ideal Gas Law, is:
[tex]\[ k = 0.00859090909090909 \][/tex]
### Summary
Given the values [tex]\( P_1 = 105 \, \text{N/m}^2 \)[/tex], [tex]\( V_1 = 0.0225 \, \text{m}^3 \)[/tex], and [tex]\( T_1 = 275 \, \text{K} \)[/tex], the constant [tex]\( k \)[/tex] from the Ideal Gas Law equation [tex]\( \frac{P_1 V_1}{T_1} \)[/tex] becomes [tex]\( 0.00859090909090909 \)[/tex]. This calculation confirms that for an ideal gas, such properties are interrelated proportionally through this constant, given the known initial conditions.
