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

Question: Consider the statements below regarding binary mixtures of substances A and B. Identify whether each statement is true or false and provide an explanation. a. The density of a mixture is 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 more than half of the moles belong to component A. d. When the molar masses of A and B are equal, the mass fraction of A and the mole fraction of A are equal. e. When the mass fractions of A and B are both 0.5, the molar mass of the mixture is the arithmetic average of the molar masses of A and B.

Step-by-step solution

Statement (a)

The statement is false. The density of a mixture is not necessarily equal to the sum of the densities of its constituents. It depends on the interactions between the constituent particles within the mixture. In general, the density of a mixture can be expressed as a weighted average of the densities of its constituents depending on their mass fractions.

Statement (b)

The statement is false. The ratio of the density of component A to the density of component B is not equal to the mass fraction of component A. The mass fraction is about comparing the mass of A and B in the mixture. To compare the densities of the two components, we need to consider both their masses and their volumes.

Statement (c)

The statement is false. When the mass fraction of component A is greater than 0.5, it implies that the mass of A is greater than the mass of B in the mixture. However, having more than half of the mass belonging to component A does not guarantee that more than half of the moles belong to component A, as molar amounts also depend on the molar masses of A and B.

Statement (d)

The statement is true. When the molar masses of A and B are equal, the mass fraction of A and the mole fraction of A will indeed be equal. This is because, in this case, the mass of A divided by the mass of B is equal to the number of moles of A divided by the number of moles of B.

Statement (e)

The statement is false. When the mass fractions of A and B are both 0.5, it doesn't necessarily mean that the molar mass of the mixture is the arithmetic average of the molar masses of A and B. Instead, the molar mass of the mixture depends on the mole fractions of A and B and their respective molar masses: $$M_\text{mixture} = x_A M_A + x_B M_B$$, where \(x_A\) and \(x_B\) represent the mole fractions of components A and B, and \(M_A\) and \(M_B\) represent the molar masses of A and B.