Question

What is the molar mass of a gas found to have a density of \(0.841 \mathrm{g} / \mathrm{L}\) at \(415 \mathrm{K}\) and 725 Torr?

Short answer

The molar mass of the gas is 29 g/mol.

Step-by-step solution

Convert the units

First, the pressure needs to be converted from Torr to atmospheres, because R's value is usually given in (L.atm)/(mol.K). The conversion factor is 1 atm = 760 Torr. Therefore, the pressure \( P = 725 \, Torr \times (1 \, atm / 760 \, Torr) = 0.953 \, atm\). Also, density d needs to be converted from g/L to mol/L. The conversion factor is 1 g/mol = 1 mol/g. Therefore, the density \( d = 0.841 \, g/L \times (1 \, mol/g) = 0.841 \, mol/L\).

Rearrange the formula to solve for M

We need to rearrange the equation derived from the ideal gas law for density to solve for molar mass M. The equation becomes \( M = dRT / P \).

Substitute the values

Plug in the values of d, R, T, and P into the rearranged equation. Using R = 0.0821 L.atm/(mol.K), \( M = 0.841 \, mol/L \times 0.0821 \, L.atm/(mol.K) \times 415 \, K / 0.953 \, atm \).

Perform the Calculation

After substituting the values into the equation, the calculation \( M = 29 \, g/mol \) is obtained.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It's expressed by the formula \(PV = nRT\), where \(P\) represents pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. This equation allows us to predict how a gas will behave under different conditions or, as in this problem, to calculate unknowns like molar mass or density.
To use this law effectively:

• Make sure pressure is in atmospheres.
• Volume should be in liters (L).
• Temperature must be in Kelvin (K).
• Moles indicate the amount of gas you're working with.

By rearranging the equation, you can also determine the molar mass of a gas when its density, temperature, and pressure are known.

Gas Density

Gas density refers to the mass of the gas divided by its volume, usually expressed in grams per liter (g/L). It's a crucial concept as it links mass and volume, allowing you to explore properties like molar mass.
For the given problem, understanding that gas density was used in the rearranged ideal gas law equation helps streamline the calculation of molar mass.
The specific method is to:

• Use the density as part of the formula \(M = \frac{dRT}{P}\)
• Remember that the density unit must be compatible with the ideal gas constant \(R\)

This tailored approach can make tackling similar problems more straightforward.

Unit Conversion

Unit conversion is essential in scientific calculations to ensure consistency and accuracy. In this exercise, converting pressure from Torr to atmospheres is critical because the ideal gas constant \(R\) is often defined in terms of liters, atmospheres, moles, and Kelvin.
Here's a simple way to convert:

• Identify the units given in the problem.
• Locate the respective conversion factors (e.g., \(1 \, atm = 760 \, Torr\)).
• Multiply or divide the original measure by the conversion factor to get the required unit.

Consistency in units helps enhance the calculation's accuracy, allowing you to apply mathematical operations appropriately.

Pressure Conversion

Pressure conversion is a specific type of unit conversion aimed at adjusting pressure measurements so they align with the other units in an equation. For the Ideal Gas Law, pressure units must be in atmospheres because the constant \(R\) uses these units.
To convert pressure from Torr to atmospheres:

• Remember the conversion factor: \(1 \, atm = 760 \, Torr\).
• Multiply the pressure in Torr by \(\frac{1 \, atm}{760 \, Torr}\) to convert it.
• Accuracy in pressure conversion ensures the correct application of the Ideal Gas Law.

Applying these steps will allow you to accurately solve gas law problems with differing initial pressure units.

Temperature

Temperature is an integral part of gas law equations, always measured in Kelvin for thermodynamic calculations like the Ideal Gas Law. The Kelvin scale ensures there are no negative values, simplifying calculations and respecting the absolute nature of temperature.
When undertaking gas calculations:

• Convert Celsius to Kelvin using the formula \(K = °C + 273.15\).
• Make sure your temperature is always in Kelvin to align with the ideal gas equation constants.

Maintaining the right temperature unit is crucial for precision in solving problems involving gas behavior.