Question

A quantity of \(0.25 \mathrm{~g}\) of a substance when vaporized displaced \(50 \mathrm{~cm}^{3}\) of air at \(0^{\circ} \mathrm{C}\) and 1 atm. The gram molecular mass of the substance will be (a) \(50 \mathrm{~g}\) (b) \(100 \mathrm{~g}\) (c) \(112 \mathrm{~g}\) (d) \(127.5 \mathrm{~g}\)

Short answer

The gram molecular mass of the substance is approximately 100 g.

Step-by-step solution

Recall the Ideal Gas Law

The Ideal Gas Law is given by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant and T is the temperature.

Calculate the number of moles (n)

Rearrange the Ideal Gas Law to solve for n (number of moles): n = PV / RT. We know that P = 1 atm, V = 50 cm^3 = 0.05 L (1 L = 1000 cm^3), R = 0.0821 L.atm/K.mol, and T = 273 K (0 degrees centigrade in Kelvin). n = (1 atm)(0.05 L) / (0.0821 L.atm/K.mol)(273 K).

Convert the number of moles to grams

Gram molecular mass (M) is the mass of the substance divided by the number of moles. As 0.25 g of the substance were vaporized, M = Mass / n. Substitute the value of n from step 2 into this equation.

Calculate the gram molecular mass of the substance

Calculate the molecular mass (M) using the formula: M = (0.25 g) / n. Plug in the value of n from the previous step to find the molar mass. Consequently, solve for M.

Key concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of an ideal gas. Its formula, expressed as PV = nRT, provides a way to calculate one of these variables if the others are known. In this law, P stands for pressure, V represents volume, n is the number of moles of gas, R is the universal gas constant (0.0821 L.atm/K.mol for ideal gases), and T is the temperature measured in Kelvin (K).

It's important for students to note that real gases often behave very closely to ideal gases under many conditions, especially at high temperatures and low pressures. The Ideal Gas Law is also incredibly useful when dealing with gas stoichiometry problems since it ties together the physical properties of gases with the amount of substance in moles.

When working with the Ideal Gas Law, it's crucial to do any necessary unit conversions, like changing the volume from centimeters cubed (cm³) to liters (L), and temperature from Celsius to Kelvin, as it ensures that all units correspond with those of the gas constant, R. This facilitates computation and increases the accuracy of the results.

Molar Mass: From Grams to Moles

Molar mass, sometimes referred to as molecular weight, is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It serves as a bridge between the mass of a substance and the amount of substance in moles. Students often calculate molar mass to convert between the mass of a substance and the number of moles when working with chemical equations and reactions.

To find the molar mass, we need to know the number of moles and the mass of the substance in grams. The formula for calculating molar mass (M) is M = mass (in grams) / number of moles (n). This simple equation is profound as it allows students to understand the relationship between the mass they can measure and the number of moles required for stoichiometric calculations.

In the given exercise, after finding the number of moles using the Ideal Gas Law, the molar mass is found by dividing the given mass of the substance (in grams) by this calculated number of moles. Always remember, since molar mass is a property of the substance, it remains constant regardless of the quantity being measured, and it's numerically equal to the molecular weight of the substance often listed in the periodic table of elements.

Stoichiometry: Balancing Matter and Equations

Stoichiometry is the aspect of chemistry that deals with the quantitative relationships or ratios between reactants and products in a chemical reaction. It is based on the law of conservation of mass, which states that in a chemical reaction, the total mass of the reactants equals the total mass of the products.

In practice, stoichiometry involves calculations that use the molar mass of substances to convert between the mass of a substance and the moles necessary to react or be produced. It applies the coefficients from balanced chemical equations to quantify the exact amounts of substances involved. For instance, if the equation states that 2 moles of hydrogen react with 1 mole of oxygen, stoichiometry allows us to calculate how much hydrogen is needed to react with a given amount of oxygen, and vice versa.

In the problem provided, after utilizing both the Ideal Gas Law and the concept of molar mass to calculate the correct number of moles and the resulting gram molecular mass of a substance, stoichiometry principles can be employed to determine how much of this substance would react with another substance or how much product can be formed. These stoichiometric calculations are crucial in creating and interpreting chemical recipes and for scientists to predict quantities needed in reactions with precision.