Question

Phosphine gas, \(\mathrm{PH}_{3},\) is toxic when it reaches a concentration of \(7 \times 10^{-5} \mathrm{mg} / \mathrm{L} .\) To what pressure does this correspond at \(25^{\circ} \mathrm{C} ?\)

Short answer

The pressure is approximately \(5.34 \times 10^{-8} \; \text{atm}.\)

Step-by-step solution

Understanding the problem

We need to find the pressure of phosphine gas when its concentration is \(7 \times 10^{-5} \; \text{mg/L}\) at a temperature of \(25^{\circ} \text{C}\). We will use the ideal gas law, \(PV = nRT\), to find this pressure. To do so, we will first convert the concentration into moles per liter.

Convert concentration to moles per liter

Given: Concentration = \(7 \times 10^{-5} \; \text{mg/L}\).Molar mass of \(\text{PH}_3 = 31.98\; \text{g/mol}\).Convert \(7 \times 10^{-5} \; \text{mg/L}\) to \(\text{g/L}\) by dividing by 1000:\[7 \times 10^{-8} \; \text{g/L}.\]Now, convert \(\text{g/L}\) to moles per liter using the molar mass:\[\text{Moles per liter} = \frac{7 \times 10^{-8} \; \text{g/L}}{31.98 \; \text{g/mol}} \approx 2.19 \times 10^{-9} \; \text{mol/L}.\]

Use the ideal gas law

The ideal gas law equation is \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume (1 L), \(n\) is the number of moles calculated, \(R\) is the ideal gas constant \(0.0821 \; \text{L atm/mol K}\), and \(T\) is the temperature in Kelvin.Convert \(25^{\circ} \text{C}\) to Kelvin:\[T = 25 + 273.15 = 298.15 \; \text{K}.\]Plug in the values:\[P = \frac{nRT}{V} = \frac{2.19 \times 10^{-9} \; \text{mol} \times 0.0821 \; \text{L atm/mol K} \times 298.15 \; \text{K}}{1 \; \text{L}}.\]

Calculate the pressure

Using the values in the ideal gas law equation:\[P = \frac{2.19 \times 10^{-9} \times 0.0821 \times 298.15}{1} \approx 5.34 \times 10^{-8} \; \text{atm}.\]Thus, the pressure corresponding to the given concentration of phosphine gas is approximately \(5.34 \times 10^{-8} \; \text{atm}.\)

Key concepts

Pressure Calculation

When you need to find the pressure of a gas using its concentration and temperature, the Ideal Gas Law is your best friend. This law states that \( PV = nRT \), where:

• \( P \) is pressure,
• \( V \) is volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant \(0.0821 \; \text{L atm/mol K}\),
• \( T \) is temperature in Kelvin.

For pressure calculation, rearrange the formula to \( P = \frac{nRT}{V} \). By carefully inserting the number of moles, the ideal gas constant, and the temperature, you get the pressure. Remember, volume (\( V \)) is typically assumed as 1 liter if not stated otherwise. This keeps units consistent and calculation simpler.

Gas Concentration Conversion

Converting gas concentration from mass (e.g., mg/L) to moles per liter (mol/L) is crucial for using the ideal gas law. Here’s how you can do it:1. **Convert mass units:** Move from milligrams to grams by dividing by 1,000. For example, \( 7 \times 10^{-5} \; \text{mg/L} \) becomes \( 7 \times 10^{-8} \; \text{g/L} \).2. **Convert to moles:** Use the molar mass of the gas, which is the mass of one mole of that gas. Divide the grams per liter value by the molar mass (like 31.98 g/mol for phosphine gas). The resulting value, \( \frac{7 \times 10^{-8} \; \text{g/L}}{31.98 \; \text{g/mol}} \), gives you the concentration in mol/L.By correctly converting these units, you align with the moles needed in the ideal gas law and pave the way for accurate calculations.

Temperature Conversion

Temperatures in calculations, especially those involving gas laws, always need to be in Kelvin. Kelvin is the absolute temperature scale used in scientific calculations.- To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.- For instance, for a common room temperature of \(25^{\circ} \text{C}\), add 273.15 to get \(298.15 \text{K}\).This conversion is essential because Kelvin ensures there are no negative temperatures, which makes the mathematical computations more straightforward and consistent. Always double-check conversions to avoid errors in gas law calculations.