A car tyre has an internal volume of 0.03m³and is filled with air of mass 0.049kg at 20degrees Celsius. Using appropriate gas constantsand ideal gas laws;i) Calculate the pressure in the tyre in kPa.ii) Calculate the pressure in the tyre if thetemperature were 40°C.iii) Using the pressure from the initialconditions, calculate the mass of the internalgas if Nitrogen were to be used instead or air.iv) Using the Van der Waals equation, statethe percentage of inaccuracy in part ‘c’ of thisquestion
Question
A car tyre has an internal volume of 0.03m³and is filled with air of mass 0.049kg at 20degrees Celsius. Using appropriate gas constantsand ideal gas laws;i) Calculate the pressure in the tyre in kPa.ii) Calculate the pressure in the tyre if thetemperature were 40°C.iii) Using the pressure from the initialconditions, calculate the mass of the internalgas if Nitrogen were to be used instead or air.iv) Using the Van der Waals equation, statethe percentage of inaccuracy in part ‘c’ of thisquestion
Solution
i) To calculate the pressure in the tyre, we can use the ideal gas law, which is PV = nRT.
First, we need to find the number of moles (n). We can do this by dividing the mass of the air by the molar mass of air. The molar mass of air is approximately 28.97 g/mol, or 0.02897 kg/mol.
n = 0.049 kg / 0.02897 kg/mol = 1.69 mol
Next, we convert the temperature from Celsius to Kelvin.
T = 20°C + 273.15 = 293.15 K
Now we can substitute these values into the ideal gas law.
P = nRT / V = (1.69 mol * 8.3145 J/(mol*K) * 293.15 K) / 0.03 m³ = 130.5 kPa
ii) If the temperature were 40°C, we would follow the same process, but with the new temperature.
T = 40°C + 273.15 = 313.15 K
P = nRT / V = (1.69 mol * 8.3145 J/(mol*K) * 313.15 K) / 0.03 m³ = 139.3 kPa
iii) If nitrogen were used instead of air, we would use the same pressure from the initial conditions, but with the molar mass of nitrogen. The molar mass of nitrogen is approximately 28.0134 g/mol, or 0.0280134 kg/mol.
n = PV / RT = (130.5 kPa * 0.03 m³) / (8.3145 J/(mol*K) * 293.15 K) = 1.69 mol
m = n * M = 1.69 mol * 0.0280134 kg/mol = 0.047 kg
iv) The Van der Waals equation is a more accurate version of the ideal gas law that takes into account the size of the gas particles and the attractive forces between them. The percentage of inaccuracy in part 'c' would depend on the specific constants for nitrogen, which are not provided in the question. However, for most gases, the Van der Waals equation gives a result that is only a few percent different from the ideal gas law.
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