Question

The mass of an evacuated \(255-\mathrm{mL}\) flask is 143.187 \(\mathrm{g} .\) The mass of the flask filled with 267 torr of an unknown gas at \(25^{\circ} \mathrm{C}\) is 143.289 \(\mathrm{g}\) . Calculate the molar mass of the unknown gas.

Short answer

The molar mass of the unknown gas is approximately 69.672 g/mol.

Step-by-step solution

- Convert volume to liters

First, convert the volume of the flask from milliliters to liters to align with standard units for gas calculations. Use the conversion factor 1 L = 1000 mL. Volume in liters = 255 mL * (1 L / 1000 mL) = 0.255 L.

- Convert temperature to Kelvin

Since gas laws require the temperature to be in Kelvin, convert the temperature from Celsius to Kelvin. Use the formula: K = °C + 273.15. Temperature in Kelvin = 25°C + 273.15 = 298.15 K.

- Calculate the mass of the unknown gas

Subtract the mass of the empty flask from the mass of the flask filled with the gas to find the mass of the gas alone. Mass of gas = 143.289 g - 143.187 g = 0.102 g.

- Convert torr to atmospheres

Standard gas equations often require pressure to be in atmospheres. Convert the given pressure from torr to atmospheres. Use the conversion factor 1 atm = 760 torr. Pressure in atmospheres = 267 torr * (1 atm / 760 torr) = 0.3513 atm.

- Use the Ideal Gas Law

With the given conditions and the Ideal Gas Law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin, calculate the number of moles of the unknown gas. Since R (the ideal gas constant) is 0.0821 L atm/(mol K), you can rearrange the formula to solve for n (moles of gas): n = PV / (RT). n = (0.3513 atm * 0.255 L) / (0.0821 L atm/(mol K) * 298.15 K) = 0.001464 mol.

- Calculate the molar mass of the unknown gas

Finally, calculate the molar mass (M) of the unknown gas using the mass of the gas and the number of moles (M = mass / moles). Molar mass = 0.102 g / 0.001464 mol = 69.672 g/mol.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry and physics that describes the behavior of an ideal gas under various conditions. It is denoted by the formula: PV = nRT, where P stands for pressure, V for volume, n for the number of moles of gas, R for the ideal gas constant, and T for temperature in Kelvin. Understanding the Ideal Gas Law is essential for solving various gas-related problems, including finding the molar mass of a gas, as in our exercise.

When solving for one of the variables, it's important to ensure all the other units are in their proper form. As seen in the solution steps for calculating the molar mass, each element is carefully converted to match the units used in the gas constant. Comprehending the applicability of this law often provides a gateway to more advanced chemistry involving gaseous substances.

Unit Conversion

Unit conversion is a fundamental skill in science that involves changing a measurement from one unit to another. It is particularly important when dealing with various quantities in chemistry and physics. In our case, to use the Ideal Gas Law, converting units of volume from milliliters to liters and pressure from torr to atmospheres is vital.

Why Unit Conversion Matters

While seemingly trivial, accurate unit conversion ensures that your calculations are correct and the final result is precise. An incorrect unit can result in a drastically different answer, which could have serious consequences in practical applications. Units act as a language that allows scientists to communicate quantitative information accurately.

Molar Mass

Molar mass is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol). It's a bridge between the microscopic scale of atoms and molecules to the macroscopic scale that we can measure in a lab. Calculating the molar mass is a fundamental aspect of stoichiometry and is used for converting between grams and moles.

For an unknown gas, as shown in the exercise, the molar mass is found by dividing the mass of a sample of the gas by the number of moles of the gas present in the sample. This calculation can reveal the molecular weight of the gas, which helps in identifying the substance. Molar mass plays a key role in understanding the composition and reactions involving chemical substances.

Gas Constant

The gas constant, denoted by R, is a physical constant that features in the Ideal Gas Law and other equations of state. It links the amount of gas to its other properties. For example, in the Ideal Gas Law, it connects pressure, volume, and temperature to the amount of gas in moles. The value of R can vary depending on the units used for pressure and volume, but in our exercise, the value used is 0.0821 L atm/(mol K), which is suitable for pressure in atmospheres and volume in liters.

Understanding the gas constant is essential for predicting how a gas will behave under various conditions. It is a unique number that applies to all ideal gases, making it a unifying feature in the study of gas laws and behavior.