Find the mass of urea (CH₄N₂O) needed to prepare 50.0 g of a solution in water in which the mole fraction of urea is 0.0770.

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Identify the given information: the mole fraction of urea (CH4N2O) is 0.0770, and the total mass of the solution is 50.0 g.

Recall that the mole fraction of a component in a solution is the ratio of the moles of that component to the total moles of all components in the solution. Therefore, \( X_{\text{urea}} = \frac{n_{\text{urea}}}{n_{\text{urea}} + n_{\text{water}}} = 0.0770 \).

Express the moles of urea \( n_{\text{urea}} \) in terms of its mass \( m_{\text{urea}} \) and molar mass \( M_{\text{urea}} \): \( n_{\text{urea}} = \frac{m_{\text{urea}}}{M_{\text{urea}}} \). The molar mass of urea (CH4N2O) is approximately 60.06 g/mol.

Express the moles of water \( n_{\text{water}} \) in terms of its mass \( m_{\text{water}} \) and molar mass \( M_{\text{water}} \): \( n_{\text{water}} = \frac{m_{\text{water}}}{M_{\text{water}}} \). The molar mass of water (H2O) is approximately 18.02 g/mol.

Set up the equation for the mole fraction: \( 0.0770 = \frac{\frac{m_{\text{urea}}}{60.06}}{\frac{m_{\text{urea}}}{60.06} + \frac{50.0 - m_{\text{urea}}}{18.02}} \). Solve this equation for \( m_{\text{urea}} \) to find the mass of urea needed.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a specific component to the total number of moles of all components in the mixture. In this case, the mole fraction of urea indicates how much of the total solution is made up of urea, which is crucial for calculating the mass needed.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For urea (CH4N2O), the molar mass can be calculated by summing the atomic masses of its constituent elements: carbon, hydrogen, nitrogen, and oxygen. Knowing the molar mass is essential for converting between moles and grams when determining how much urea is required for the solution.

Mass Percent and Solution Preparation

Mass percent is the mass of a solute divided by the total mass of the solution, multiplied by 100. In this context, understanding how to relate the mass of urea to the total mass of the solution (50.0 g) is vital for preparing the desired solution. This involves using the mole fraction to find the number of moles of urea and then converting that to mass using the molar mass.

Find the mass of urea (CH4N2O) needed to prepare 50.0 g of a solution in water in which the mole fraction of urea is 0.0770.