Question

Three kilograms of water in a container have a pressure of \(100 \mathrm{kPa}\) and temperature of \(150^{\circ} \mathrm{C}\). What is the volume of this container?

Short answer

Answer: The volume of the container is approximately 5.719 m³.

Step-by-step solution

Convert temperature to Kelvin

First, we need to convert the temperature from Celsius to Kelvin. To do this, we add 273.15 to the given temperature: \(150^{\circ} \mathrm{C} + 273.15 = 423.15\mathrm{K}\).

Calculate the number of moles

Next, we need to find the number of moles of water (n) in the container. To do that, we use the molar mass of water, which is \(18.015\mathrm{g/mol}\). First, convert the mass from kg to grams: \(3\mathrm{kg} * 1000 = 3000\mathrm{g}\). Now, divide the mass by the molar mass: \(n = \frac{3000\mathrm{g}}{18.015\mathrm{g/mol}} \approx 166.53\mathrm{mol}\).

Use the Ideal Gas Law to find the volume

Now, we can use the Ideal Gas Law equation: \(PV = nRT\). We know the pressure, P = \(100\mathrm{kPa}\) (note that \(1\mathrm{kPa} = 1000\mathrm{Pa}\)), the number of moles, n = \(166.53\mathrm{mol}\), the gas constant, \(R = 8.314\mathrm{J/(mol\cdot K)}\), and the temperature, T = \(423.15\mathrm{K}\). Plug in the values and solve for V: \(V = \frac{nRT}{P} = \frac{166.53\mathrm{mol} \times 8.314\mathrm{J/(mol\cdot K)} \times 423.15\mathrm{K}}{100000\mathrm{Pa}}\)

Calculate the volume

Performing the calculation, we find that the volume of the container is approximately \(5.719\mathrm{m^3}\). Therefore, the volume of the container is \(5.719\mathrm{m^3}\).