Question

Mark these statements as being True or False for a binary mixture of substances \(A\) and \(B\). (a) The density of a mixture is always equal to the sum of the densities of its constituents. (b) The ratio of the density of component \(A\) to the density of component \(B\) is equal to the mass fraction of component \(A\). (c) If the mass fraction of component \(A\) is greater than \(0.5\), then at least half of the moles of the mixture are component \(A\). (d) If the molar masses of \(A\) and \(B\) are equal to each other, then the mass fraction of \(A\) will be equal to the mole fraction of \(A\). (e) If the mass fractions of \(A\) and \(B\) are both \(0.5\), then the molar mass of the mixture is simply the arithmetic average of the molar masses of \(A\) and \(B\).

Short answer

In summary: a) The density of a mixture is always equal to the sum of the densities of its constituents. - False b) The ratio of the density of component A to the density of component B is equal to the mass fraction of component A. - False c) If the mass fraction of component A is greater than 0.5, then at least half of the moles of the mixture are component A. - False d) If the molar masses of A and B are equal to each other, then the mass fraction of A will be equal to the mole fraction of A. - True e) If the mass fractions of A and B are both 0.5, then the molar mass of the mixture is simply the arithmetic average of the molar masses of A and B. - True

Step-by-step solution

Statement (a): Maintaining True or False

The density of a mixture is always equal to the sum of the densities of its constituents. The total density of a binary mixture is given by the relation: \(\rho_{mixture} = \frac{m_{A} + m_{B}}{V_{mixture}}\) However, the volume of the mixture may not be equal to the sum of the volumes of its constituents. Therefore, this statement is False.

Statement (b): Maintaining True or False

The ratio of the density of component \(A\) to the density of component \(B\) is equal to the mass fraction of component A. Mass fraction of component A (\(\chi_A\)) is defined as the mass of A divided by the total mass of the mixture: \(\chi_A = \frac{m_A}{m_A + m_B}\) But the given statement talks about the ratio of densities, which has no direct relation to the mass fraction. Therefore, the statement is False.

Statement (c): Maintaining True or False

If the mass fraction of component \(A\) is greater than \(0.5\), then at least half of the moles of the mixture are component A. Mole fraction of component A (\(x_A\)) is defined as the moles of A divided by the total moles of the mixture: \(x_A = \frac{n_A}{n_A + n_B}\) However, mass fraction and mole fraction have different units and are not directly related. The mass fraction tells us the fraction of mass in terms of substance A, while the mole fraction tells us the number of moles of substance A present. The statement makes an inaccurate comparison, so it is False.

Statement (d): Maintaining True or False

If the molar masses of \(A\) and \(B\) are equal to each other, then the mass fraction of \(A\) will be equal to the mole fraction of \(A\). Let's denote the molar masses of A and B as \(M_A\) and \(M_B\). Since \(M_A = M_B\), we can write the relation: \(\chi_A = \frac{m_A}{m_A + m_B} = \frac{n_A M_A}{n_A M_A + n_B M_A}\) Since \(M_A\) is equal to \(M_B\), we can cancel out the molar masses: \(\chi_A = \frac{n_A}{n_A + n_B} = x_A\) Thus, the statement is True.

Statement (e): Maintaining True or False

If the mass fractions of \(A\) and \(B\) are both \(0.5\), then the molar mass of the mixture is simply the arithmetic average of the molar masses of \(A\) and \(B\). For a binary mixture, the mass fraction of the two components always sums up to 1: \(\chi_A + \chi_B = 1\) Given mass fractions of A and B are both 0.5, so \(\chi_A = \chi_B = 0.5\) Now, let's find the molar mass of the mixture: \(M_{mixture} = \chi_A M_A + \chi_B M_B = 0.5 M_A + 0.5 M_B\) This is the arithmetic average of the molar masses of A and B, so the statement is True.

Key concepts

Binary Mixtures

A binary mixture is a combination of two different substances or components. These mixtures are a major part of studying heat and mass transfer due to their practical applications. They help us understand how different substances interact, mix and influence each other's properties.
In binary mixtures:

• Each component can have distinct properties, such as density or molar mass.
• The mixture's overall properties depend on the interaction between its components.

Understanding how different components behave in a mixture is essential for fields like material science, chemical engineering, and environmental studies.

Density of Mixtures

The density of a mixture does not simply equate to the sum of the densities of its individual components. Instead, it is calculated by considering the total mass over the total volume of the mixture. The relationship is expressed as:\[\rho_{mixture} = \frac{m_{A} + m_{B}}{V_{mixture}}\]Here:

• \(m_{A}\) and \(m_{B}\) are the masses of components A and B.
• \(V_{mixture}\) is the total volume of the mixture, which is not necessarily the sum of the volumes of its parts due to possible volume contraction or expansion when mixed.

This calculation showcases the importance of understanding how combining substances affects the total volume, and consequently, the density.

Mass Fraction and Mole Fraction

Mass fraction and mole fraction are two distinct ways to express the composition of a mixture.

• Mass Fraction (\(\chi_A\)) is defined as the mass of a component divided by the total mass of the mixture: \[\chi_A = \frac{m_A}{m_A + m_B}\] It expresses the percentage of the mixture that is made up of that component by mass.
• Mole Fraction (\(x_A\)) is the ratio of the moles of a component to the total moles in the mixture: \[x_A = \frac{n_A}{n_A + n_B}\] This gives the proportion of the substance in terms of moles.

It's important to note that mass fractions and mole fractions are different and not interchangeable, as they represent the mixture composition in different terms.

Molar Mass Calculations

Molar mass calculations are used to determine the overall molar mass of a binary mixture. When the mass fractions of components are considered, the molar mass of the mixture is given by:\[M_{mixture} = \chi_A M_A + \chi_B M_B\]Where:

• \(\chi_A\) and \(\chi_B\) are the mass fractions of components A and B.
• \(M_A\) and \(M_B\) are their respective molar masses.

If the mass fractions are equal, as in the case when both are 0.5, this formula simplifies to calculating the arithmetic mean of the two molar masses. This provides a clear understanding of how different compositions in a mixture affect its overall molar mass.