To solve this problem, we need to utilize the relationship given: the volume of gas [tex]\( (V \, \text{cm}^3) \)[/tex] varies directly as its absolute temperature [tex]\( (T \, \text{K}) \)[/tex] and inversely as its pressure [tex]\( (P \, \text{kN/m}^2) \)[/tex].
This relationship can be expressed mathematically as:
[tex]\[ V = k \frac{T}{P} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
First, let's determine the constant [tex]\( k \)[/tex] using the initial conditions provided:
- Initial volume, [tex]\( V_0 = 75 \, \text{cm}^3 \)[/tex]
- Initial pressure, [tex]\( P_0 = 40 \, \text{kN/m}^2 \)[/tex]
- Initial temperature, [tex]\( T_0 = 50 \, \text{K} \)[/tex]
Using the formula:
[tex]\[ k = \frac{V_0 \cdot P_0}{T_0} \][/tex]
Substitute the known values:
[tex]\[ k = \frac{75 \, \text{cm}^3 \cdot 40 \, \text{kN/m}^2}{50 \, \text{K}} \][/tex]
Now, we need to calculate the new volume and pressure in the subsequent parts:
### Part (i):
Find the volume collected at a pressure of [tex]\( 80 \, \text{kN/m}^2 \)[/tex] and temperature of [tex]\( 25 \, \text{K} \)[/tex].
Using the given relationship:
[tex]\[ V_1 = k \frac{T_1}{P_1} \][/tex]
Here:
- [tex]\( P_1 = 80 \, \text{kN/m}^2 \)[/tex]
- [tex]\( T_1 = 25 \, \text{K} \)[/tex]
Substitute the values:
[tex]\[ V_1 = \frac{k \cdot 25 \, \text{K}}{80 \, \text{kN/m}^2} \][/tex]
Simplifying the expression with the known value of [tex]\( k \)[/tex], we find:
[tex]\[ V_1 = 18.75 \, \text{cm}^3 \][/tex]
### Part (ii):
Find the pressure when the volume is [tex]\( 200 \, \text{cm}^3 \)[/tex] at a temperature of [tex]\( 100 \, \text{K} \)[/tex].
Using the same relationship rearranged for pressure:
[tex]\[ P_2 = k \frac{T_2}{V_2} \][/tex]
Here:
- [tex]\( V_2 = 200 \, \text{cm}^3 \)[/tex]
- [tex]\( T_2 = 100 \, \text{K} \)[/tex]
Substitute the values:
[tex]\[ P_2 = \frac{k \cdot 100 \, \text{K}}{200 \, \text{cm}^3} \][/tex]
Simplifying the expression with the known value of [tex]\( k \)[/tex], we find:
[tex]\[ P_2 = 30 \, \text{kN/m}^2 \][/tex]
Therefore, the solutions to the problem are:
(i) The volume collected at a pressure of [tex]\( 80 \, \text{kN/m}^2 \)[/tex] and temperature of [tex]\( 25 \, \text{K} \)[/tex] is [tex]\( 18.75 \, \text{cm}^3 \)[/tex].
(ii) The pressure when the volume is [tex]\( 200 \, \text{cm}^3 \)[/tex] at a temperature of [tex]\( 100 \, \text{K} \)[/tex] is [tex]\( 30 \, \text{kN/m}^2 \)[/tex].
