Question

During a hot summer day at the beach when the air temperature is \(30^{\circ} \mathrm{C},\) someone claims the vapor pressure in the air to be \(5.2 \mathrm{kPa}\). Is this claim reasonable?

Short answer

Answer: Yes, the claim is reasonable, as the given vapor pressure of \(39.0032\,\mathrm{torr}\) is less than the calculated saturation vapor pressure of \(42.46\,\mathrm{torr}\) at \(30^{\circ}\mathrm{C}\).

Step-by-step solution

Understand Antoine's Equation

Antoine's equation is a semi-empirical relationship between vapor pressure and temperature, used to estimate the saturation vapor pressure. The equation is given by: \[\log_{10}(P) = A - \frac{B}{(C + T)}\] where \(P\) is the saturation vapor pressure (in torr), \(T\) is the temperature (in Celsius), and \(A, B, C\) are Antoine coefficients that depend on the substance of interest in this case, water. For water (temperature in Celsius and vapor pressure in torr): \(A = 8.07131\) \(B = 1730.63\) \(C = 233.426\)

Convert kPa to torr

First, convert the given vapor pressure in kPa to torr, since the Antoine coefficients are based on the temperature in °C and pressure in torr. Use the conversion factor: \(1\,\mathrm{kPa} = 7.50062\,\mathrm{torr}\). \[P_{given} = 5.2\,\mathrm{kPa}\times \frac{7.50062\,\mathrm{torr}}{1\,\mathrm{kPa}} = 39.0032\,\mathrm{torr}\]

Calculate saturation vapor pressure using Antoine's Equation

Now plug the temperature and Antoine's coefficients in the equation to calculate the water saturation vapor pressure at \(30^{\circ}\mathrm{C}\): \[\log_{10}(P) = 8.07131 - \frac{1730.63}{233.426 + 30}\] Solve this equation for \(P\): \[P = 10^{log_{10}(P)}\] \[P_{saturation} = 10^{(8.07131 - \frac{1730.63}{(233.426 + 30)})} \approx 42.46\,\mathrm{torr}\]

Compare given vapor pressure with saturation vapor pressure

Now compare the given vapor pressure \(39.0032\,\mathrm{torr}\) with the calculated saturation vapor pressure of \(42.46\,\mathrm{torr}\). Since the given vapor pressure (\(39.0032\,\mathrm{torr}\)) is less than the saturation vapor pressure (\(42.46\,\mathrm{torr}\)), the claim is reasonable.

Key concepts

Vapor Pressure

Vapor pressure is an essential concept in understanding the behavior of liquids and gases in various conditions. It is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The vapor pressure of a substance indicates its volatility—the tendency of a substance to vaporize. At any given temperature, a liquid will produce vapor until the pressure of that vapor reaches a certain level, this level being the vapor pressure at that temperature. As the temperature increases, the vapor pressure also increases since molecules have more energy to escape from the liquid or solid phase into the gas phase. This relationship between temperature and vapor pressure is crucial in many industrial and scientific applications, such as distillation, weather prediction, and even the storage of volatile goods.

Saturation Vapor Pressure

Saturation vapor pressure is a specific type of vapor pressure—it is the maximum pressure achieved by the vapor when the liquid (or solid) and vapor are in dynamic equilibrium. At this point, the rate at which molecules evaporate from the liquid equals the rate at which they condense back into it. The critical aspect of saturation vapor pressure is that it depends solely on the temperature; for a given substance, higher temperatures will correspond to higher saturation vapor pressures. Understanding saturation vapor pressure is vital in weather science, where it helps predict the formation of dew, clouds, and fog. It is also important in understanding the concept of relative humidity, which compares the current vapor pressure to the saturation vapor pressure to determine how 'full' the air is with water vapor.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, energy, and the relationships between them. It's a foundational science that explains how energy is converted from one form to another and how it affects matter. The principles of thermodynamics are applied in a wide array of scientific disciplines, including chemistry and engineering. In the context of Antoine's Equation and vapor pressure, thermodynamics explains how energy affects the phase transitions of substances. Particularly the first and second laws of thermodynamics are at play, dictating that energy cannot be created or destroyed and that entropy - a measure of molecular disorder - increases over time, influencing the transition of water from liquid to gas phase, which in turn is reflected in its vapor pressure.

Temperature Conversion

Temperature conversion is a necessary tool in many scientific calculations since various regions and scientific disciplines use different temperature scales. The most common scales are Celsius, Fahrenheit, and Kelvin. In Antoine's Equation, we typically use the Celsius scale when dealing with substances like water. However, conversions may be necessary when working with measurements that have been made using different units or when applying scientific formulas. For example, converting from Celsius to Kelvin would involve adding 273.15 to the Celsius value. In the case of vapor pressure calculations, understanding temperature conversions is fundamental, as Antoine's Equation requires temperature values in Celsius, and errors in conversion can result in inaccurate vapor pressure estimates.