Question

The vapor pressure of \(\mathrm{Hg}(1)\) as a function of temperature is \(\log P(\mathrm{mmHg})=(-0.05223 a / T)+b,\) where \(a=61,960\) and \(b=8.118 ; T\) is the Kelvin temperature. Show that at \(25^{\circ} \mathrm{C}\), the concentration of \(\mathrm{Hg}(\mathrm{g})\) in equilibrium with Hg(l) greatly exceeds the maximum permissible level of \(0.05 \mathrm{mg} \mathrm{Hg} / \mathrm{m}^{3}\) air.

Short answer

By applying the temperature dependence of the vapor pressure and subsequent conversion to the concentration, it can be shown that the concentration of \(Hg(g)\) at \(25^\circ C\) indeed exceeding the maximum permissible level of \(0.05 mg/m^3\) significantly.

Step-by-step solution

Convert the temperature to Kelvin

The temperature is provided in Celsius but the pressure-temperature relationship works with Kelvin temperatures. Therefore, convert \(25^\circ C\) to Kelvin: \(K = C + 273.15 \rightarrow T = 25 + 273.15 = 298.15 K\)

Substitute temperature into given equation

Substitute the value of \(T\) from step 1 into the given logarithmic relationship for pressure: \(\log P = (-0.05223 \cdot 61960 / T) + 8.118\). When inserting \(T = 298.15 K\) into the equation, the calculation will give the vapor pressure of Hg(l) at the given temperature.

Conversion from log base 10 to real number

Use the 10 exponent to convert the logarithmic pressure: \(P = 10^{\log P}\). This will give the vapor pressure of Hg(l) in mmHg.

Convert pressure to concentration

Use the ideal gas law to convert this pressure to a concentration. The ideal gas law is \(PV = nRT\) where P is pressure, V is volume, n is number of moles, R is the gas constant, and T is temperature. We are looking for concentration (n/V), so rearrange to \(C = P / (RT)\). Convert pressure from mmHg to atm by dividing by 760 as R expects pressure in atm. Convert concentration to \(mg/m^3\) for comparison to permissible level.

Compare to permissible level

Compare determined concentration to permissible level of \(0.05 mg/m^3\). If determined concentration is higher, this verifies that the concentration of \(Hg(g)\) in equilibrium with \(Hg(l)\) greatly exceeds the maximum permissible level.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry where the rate of the forward reaction equals the rate of the backward reaction, resulting in no net change in the concentrations of the reactants and products over time. This state is dynamic, meaning that the reactions continue to occur, but maintain a balance. In the context of vapor pressure, when a liquid (like mercury, Hg(l)) is in a closed container, it will evaporate, forming a gas phase. The gas molecules can also condense back into the liquid. At equilibrium, the rate of evaporation equals the rate of condensation.

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature. For mercury, the vapor pressure indicates how many mercury atoms have evaporated and are in the gas phase at a specific temperature. The exercise requires the quantification of mercury vapor concentration at a certain temperature to determine if it's above a safe threshold. Understanding equilibrium helps to appreciate why changing conditions like temperature can affect the concentration of hazardous mercury vapor in the air.

Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and amount of an ideal gas through the formula: \( PV = nRT \). Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This law assumes that gases consist of widely spaced particles that interact minimally, which is a simplified model that works remarkably well for many gases under typical conditions.

In our exercise, the ideal gas law is rearranged to solve for concentration (n/V) as a function of pressure and temperature: \( C = P / (RT) \). When the vapor pressure of mercury is obtained, it can be converted into a concentration using this law, assuming mercury behaves as an ideal gas. This assumption allows us to relate a quantity that is easily measured, pressure, to the crucial figure needed – the concentration of Hg(g) in the air, to determine if it exceeds health standards.

Conversion of Temperature Units

Understanding how to convert temperature units is crucial in scientific calculations. The given problem involves the conversion of Celsius to Kelvin, which is required because the formula for vapor pressure of mercury uses Kelvin. To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature: \( K = C + 273.15 \).

The temperature in Kelvin is crucial as it's an absolute temperature scale used in scientific equations like the ideal gas law and the vapor pressure equation for the consistency of units. These calculations require precision, and without the correct conversion of temperature units, we could end up with erroneous results. Always remember when dealing with thermodynamic properties or reaction rates, Kelvin is the standard temperature unit used to ensure that calculated values are accurate and reflective of true thermal energy levels.