Question

A certain hydrate has the formula \(\mathrm{MgSO}_{4} \cdot x \mathrm{H}_{2} \mathrm{O} .\) A quantity of \(54.2 \mathrm{~g}\) of the compound is heated in an oven to drive off the water. If the steam generated exerts a pressure of 24.8 atm in a 2.00-L container at \(120^{\circ} \mathrm{C}\), calculate \(x\)

Short answer

x = 7.

Step-by-step solution

Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \] Thus, the temperature in Kelvin is \( 120 + 273.15 = 393.15 \) K.

Use Ideal Gas Law for Water Vapor

Apply the Ideal Gas Law \( PV = nRT \) to find the moles of water vapor. Rearrange for \( n \): \[ n = \frac{PV}{RT} \] Given \( P = 24.8 \) atm, \( V = 2.00 \) L, and \( T = 393.15 \) K, with the ideal gas constant \( R = 0.0821 \) L·atm/mol·K, calculate \( n \): \[ n = \frac{24.8 \times 2.00}{0.0821 \times 393.15} \approx 1.544 \, \text{mol} \]

Calculate Mass of Water Lost

The mass of 1 mole of water is 18.02 g/mol. Multiply the moles of water lost by its molar mass: \[ \text{mass of } \mathrm{H}_2\mathrm{O} = 1.544 \times 18.02 = 27.81 \text{ g} \]

Find Mass of Anhydrous MgSO4

Subtract the mass of the water lost from the total mass of the hydrate to find the mass of anhydrous \( \mathrm{MgSO}_4 \): \[ \text{mass of anhydrous } \mathrm{MgSO}_4 = 54.2 - 27.81 = 26.39 \text{ g} \]

Calculate Moles of MgSO4

Find the moles of anhydrous \( \mathrm{MgSO}_4 \) using its molar mass (120.37 g/mol): \[ \text{moles of } \mathrm{MgSO}_4 = \frac{26.39}{120.37} \approx 0.2194 \text{ mol} \]

Determine x in the Hydrate Formula

The moles of \( \mathrm{H}_2\mathrm{O} \) relates to \( x \) in the hydrate formula using moles of anhydrous \( \mathrm{MgSO}_4 \). The ratio gives \( x \): \[ x = \frac{1.544}{0.2194} \approx 7.04 \] Rounding to the nearest whole number, \( x = 7 \).

Key concepts

Ideal Gas Law

The ideal gas law is a key concept used in chemistry to relate the properties of gases. It is expressed as \( PV = nRT \), where:

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

This law helps in calculating the amount of gas produced in a reaction based on the conditions provided. Converting temperatures from Celsius to Kelvin is crucial because gas calculations are always done in Kelvin. In the exercise, the ideal gas law is used to find the moles of water vapor evolved when water is removed from the hydrate.

Moles of Water

Understanding moles is essential for interpreting chemical reactions accurately. A mole is a standard unit in chemistry that provides a way to count entities like atoms and molecules. For water in chemical equations, knowing the moles helps to understand proportions and balance. In the exercise, water vapor is produced from the heating of a hydrate. By using the ideal gas law, we can find out how many moles of water vapor are produced from the measured pressure, volume, and temperature conditions: - Pressure is given as 24.8 atm, - Volume is given as 2.00 L, - Temperature must be converted to Kelvin to accurately use in our formula.

Molar Mass

Molar mass is the mass of one mole of a substance, usually measured in grams/mol. It's crucial for converting between moles and grams. For water, the molar mass is approximately 18.02 g/mol. In the exercise, once the moles of water lost are calculated using the ideal gas law, the molar mass is utilized to determine how many grams of water were actually removed from the hydrate. Performing this conversion allows you to figure out what portion of the original compound was made up of water.

Chemical Reactions

Chemical reactions involve rearrangement of atoms and changes in substance composition. Hydrates are compounds that contain water molecules integrated into their structure. During heating, these water molecules are often released as water vapor, and this is what's analyzed in the problem.The goal in these types of exercises is to determine the number of water molecules per formula unit (or how many water molecules are attached to the main compound). The idea is to measure the hydrate before and after heating to find out how much water was expelled, which helps in calculating the integer \(x\) in the hydrate formula \(\mathrm{MgSO_4} \cdot x \mathrm{H_2O}\). With the number of moles of water and \(\mathrm{MgSO_4}\) known, the ratio helps to pinpoint the value of \(x\), giving insight into the composition of the compound.