To calculate the difference in volume of air in the shop when the temperature changes from 20.0 Celsius to 0 Celsius, we can use the ideal gas law.
The ideal gas law states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature in Kelvin.
First, we need to convert the temperatures from Celsius to Kelvin:
20.0 Celsius + 273.15 = 293.15 Kelvin
0 Celsius + 273.15 = 273.15 Kelvin
Next, we assume that the pressure and the number of moles of gas remain constant. Therefore, we can rearrange the ideal gas law equation to solve for the volume ratio:
V1 / V2 = T1 / T2
V1 = 500 m^3 (volume at 20.0 Celsius)
V2 = Unknown volume at 0 Celsius
T1 = 293.15 Kelvin (temperature at 20.0 Celsius)
T2 = 273.15 Kelvin (temperature at 0 Celsius)
Plugging in the values, we have:
500 m^3 / V2 = 293.15 K / 273.15 K
Now we can solve for V2:
V2 = (500 m^3 * 273.15 K) / 293.15 K
Calculating this equation gives us the volume at 0 Celsius:
V2 ≈ 467.21 m^3
Therefore, the difference in volume when the temperature changes from 20.0 Celsius to 0 Celsius is approximately:
500 m^3 - 467.21 m^3 = 32.79 m^3