A cylinder contains a gas. Given: Molar mass of gas M = 0.024 kg/mol, Pressure P = 1.40 × 105 N/m2, Temperature Tc = 78 °C, Cylinder height h = 7.2 cm, Cylinder diameter d = 2.5 cm. What mass of gas i

The radius of the cylinder is $r = \frac{d}{2} = \frac{0.025 \, \text{m}}{2} = 0.0125 \, \text{m}$.
The volume $V$ of the cylinder is:

$$V = \pi r^2 h = \pi (0.0125 \, \text{m})^2 (0.072 \, \text{m})$$ $$V = \pi (0.00015625 \, \text{m}^2) (0.072 \, \text{m}) = 0.0000353429 \, \text{m}^3$$

Convert the temperature from Celsius to Kelvin:

$$T = T_C + 273.15 = 78 \, ^\circ \text{C} + 273.15 = 351.15 \, \text{K}$$

Part (a) i) What mass of gas is present in the container?

The ideal gas law is $PV = nRT$. Rearranging for $n$:

$$n = \frac{PV}{RT}$$ $$n = \frac{(1.40 \times 10^5 \, \text{N/m}^2) \times (0.0000353429 \, \text{m}^3)}{(8.314 \, \text{J/(mol} \cdot \text{K)}) \times (351.15 \, \text{K})}$$ $$n = \frac{4.948006}{2919.9861} = 0.0016945 \, \text{mol}$$

The mass of the gas is the number of moles multiplied by the molar mass:

$$m = n \times M$$ $$m = 0.0016945 \, \text{mol} \times 0.024 \, \text{kg/mol}$$ $$m = 0.000040668 \, \text{kg}$$

Rounding to three significant figures:

$$\boxed{m = 4.07 \times 10^{-5} \, \text{kg}}$$

Part (a) ii) How many molecules of the gas are present in the container?

The number of molecules is the number of moles multiplied by Avogadro's number:

$$N = n \times N_A$$ $$N = 0.0016945 \, \text{mol} \times 6.022 \times 10^{23} \, \text{mol}^{-1}$$ $$N = 1.0203 \times 10^{21} \, \text{molecules}$$

Rounding to three significant figures:

$$\boxed{N = 1.02 \times 10^{21} \, \text{molecules}}$$

Part (b) Calculate the average kinetic energy of the gas molecules.

For an ideal gas, the average translational kinetic energy per molecule is given by:

$$KE_{avg} = \frac{3}{2} kT$$ $$KE_{avg} = \frac{3}{2} \times (1.38 \times 10^{-23} \, \text{J/K}) \times (351.15 \, \text{K})$$ $$KE_{avg} = 1.5 \times 1.38 \times 10^{-23} \times 351.15 \, \text{J}$$ $$KE_{avg} = 7.2798 \times 10^{-21} \, \text{J}$$

Rounding to three significant figures:

$$\boxed{KE_{avg} = 7.28 \times 10^{-21} \, \text{J}}$$