Question

What is the vapor pressure of water at \(100^{\circ} \mathrm{C} ?\) How do you know?

Short answer

The vapor pressure of water at 100°C is approximately 758.3 mmHg. This value is obtained using the Antoine equation and the given constants for water, which relates the vapor pressure to temperature. At this temperature, water molecules have enough energy to escape into the vapor phase, and the vapor pressure is close to the standard atmospheric pressure, which is why water boils at 100°C.

Step-by-step solution

Understanding Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phase (liquid or solid) at a given temperature in a closed system. In simple terms, it is the pressure at which liquid and vapor phase can coexist at a given temperature. The vapor pressure of a substance depends on its temperature: as the temperature increases, the vapor pressure increases.

Using a Vapor Pressure Table or Formula

To find the vapor pressure of water at 100°C (373.15 K), we can either consult a vapor pressure table for water or use the Antoine equation, which is a common empirical formula used to estimate vapor pressure. In this solution, we will use the Antoine Equation, given as: \(log_{10}P = A - \frac{B}{C + T}\) where: - P is the vapor pressure (in mmHg), - T is the temperature (in Celsius), - A, B, and C are constants specific to each substance. For water, A=8.07131, B=1730.63, and C=233.426.

Calculation

Now, we will use the Antoine equation with the given constants for water and the temperature (100°C) to calculate the vapor pressure: \(log_{10}P = 8.07131 - \frac{1730.63}{233.426 + 100}\) \(log_{10}P = 8.07131 - \frac{1730.63}{333.426}\) \(log_{10}P = 8.07131 - 5.1958\) \(log_{10}P = 2.87551\) Now, to find the vapor pressure, we raise 10 to the power of the result: \(P = 10^{2.87551}\) \(P \approx 758.3\ mmHg\)

Interpreting the Result

The vapor pressure of water at 100°C is approximately 758.3 mmHg. This means that at 100°C, water molecules in the liquid phase have enough energy to escape into the vapor phase, and the pressure exerted by the water vapor is 758.3 mmHg. This value also confirms our knowledge that water boils at 100°C at standard atmospheric pressure (760 mmHg), as the vapor pressure of water is very close to the atmospheric pressure at this temperature.