Question

A \(113-\mathrm{mL}\) gas sample has a mass of 0.171 \(\mathrm{g}\) at a pressure of 721 \(\mathrm{mm} \mathrm{Hg}\) and a temperature of \(32^{\circ} \mathrm{C} .\) What is the molar mass of the gas?

Short answer

The molar mass of the gas is approximately 40.56 g/mol.

Step-by-step solution

Convert temperature from Celsius to Kelvin

Temperature must be in Kelvin for gas law calculations. Convert the temperature from Celsius to Kelvin using the formula: \( T(K) = T(^\text{\textdegree}C) + 273.15 \). In this case, \( T = 32^\text{\textdegree}C + 273.15 = 305.15 K \).

Convert pressure from mmHg to atm

Pressure must be in atmospheres (atm) for the Ideal Gas Law. Convert pressure from mmHg to atm using the conversion factor: \( 1 atm = 760 mmHg \). Thus, \( P = \frac{721 mmHg}{760 mmHg/atm} = 0.94868 atm \).

Apply the Ideal Gas Law to find moles of gas

Use the Ideal Gas Law equation, \( PV = nRT \), to find the number of moles \( n \). The gas constant \( R = 0.0821 \frac{L\cdot atm}{mol\cdot K} \). Solve for \( n \) in the equation \( n = \frac{PV}{RT} \). With the given volumes in liters, \( n = \frac{(0.94868 atm) \cdot (0.113 L)}{(0.0821 \frac{L\cdot atm}{mol\cdot K}) \cdot (305.15 K)} = 4.2145 \times 10^{-3} mol \).

Calculate the molar mass of the gas

The molar mass (M) of the gas can be found using the mass of the gas (m) and the number of moles (n). Use the formula \( M = \frac{m}{n} \). The mass of the gas sample is 0.171 g, so \( M = \frac{0.171 g}{4.2145 \times 10^{-3} mol} = 40.56 \frac{g}{mol} \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas through the formula \[\[\begin{align*} PV = nRT \end{align*}\]\]where

• P is the pressure of the gas,
• V is the volume of the gas,
• n is the number of moles of the gas,
• R is the ideal gas constant, and
• T is the absolute temperature of the gas in Kelvin.

This law is particularly useful for understanding how a gas behaves under different conditions of temperature and pressure. By manipulating the equation, one can solve for any one of the variables if the others are known.In relating this to the exercise, the Ideal Gas Law allows us to determine the number of moles of the gas sample when the pressure, volume, and temperature are given. Knowing the number of moles is crucial for finding the molar mass of the unknown gas.

Temperature Conversion

Temperature conversion is an essential part of working with gas laws since the laws require temperature to be in Kelvin (K). To convert degrees Celsius (°C) to Kelvin, the formula is: \[\[\begin{align*} T(K) = T(^\circ C) + 273.15 \end{align*}\]\]This conversion is necessary because Kelvin is the absolute temperature scale used in scientific measurements and it sets the zero point at absolute zero, the theoretical point where molecular motion ceases.

Practical Usage in Our Example

In the given exercise, the initial temperature is provided in Celsius. By adding 273.15, we adjust the scale to Kelvin, which is required for using in the Ideal Gas Law. This ensures accuracy in calculations relating to the behavior of gases.

Pressure Conversion

Pressure conversion is commonly needed in gas law problems since standard laboratory measurements and the Ideal Gas Law may use different units for pressure. In most chemistry problems, pressure is converted to atmospheres (atm), because it is the standard unit used in the Ideal Gas Law equation. Here is the conversion factor between atmospheres and millimeters of mercury (mmHg): \[\[\begin{align*} 1 \, atm = 760 \, mmHg \end{align*}\]\]

Implementation in the Exercise

For our example, the pressure was initially measured in mmHg, which we then converted to atm using the conversion factor mentioned above. By converting to atm, we can plug this value directly into the Ideal Gas Law without the need for additional conversions, simplifying our calculations and maintaining consistency in units.

Gas Constant R

The gas constant, denoted as R, is a proportionality constant that appears in the Ideal Gas Law. Its value depends on the units used for pressure, volume, and temperature. For pressure in atmospheres (atm) and volume in liters (L), the value of R is: \[\[\begin{align*} R = 0.0821 \, \frac{L\cdot atm}{mol\cdot K} \end{align*}\]\]This value is pivotal in calculations involving the Ideal Gas Law because it bridges the macroscopic measurements of gases (pressure, volume, and temperature) with the microscopic property of mole quantity.

Role in the Solution

By using the appropriate value for R, we can accurately calculate the number of moles of the gas in our exercise. The constant R effectively relates the physical conditions of the gas sample to its amount in terms of moles, enabling us to derive the molar mass of the gas after determining the number of moles present.