Answer
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Hint: Mole fraction is defined as the number of moles of a particular component in a mixture divided by the total number of moles in that solute-solvent mixture. Ratio of given mass to molar mass gives the value of number of moles.
Complete step-by-step solution:
Mole fraction is the ratio of moles of one substance in a solution mixture to the total number of moles of all substances in the same mixture. For a mixture of two substances, A and B, we can write the mole fractions of each as follows:
\[mole{\text{ }}fraction{\text{ }}of{\text{ }}A = \dfrac{{{n_A}}}{{{n_A} + {n_B}}}\]
We can calculate number of moles of each substance using the below formula:
\[moles = \dfrac{{given{\text{ }}mass}}{{molar{\text{ }}mass}}\]
Number of moles of water are
\[moles = \dfrac{{100}}{{18}} = 5.55moles\]
We are provided that the mole fraction of ethyl alcohol is 0.2 and molecular weight of ethyl alcohol is 46 grams, putting these values in mole fraction formula, we get
\[0.2 = \dfrac{{{m_x}/46}}{{{m_x}/46 + 5.55}}\]
Solving this equation, we get the value of \[{m_x}\]= 64.4 grams.
Therefore, 64.4 grams of ethyl alcohol must be mixed with 100ml of water for the mole fraction (X) of ethyl alcohol to be 0.2.
Hence, the correct option is (D).
Note: The sum of mole fraction of all the components in the solution mixture is always equal to unity. Since it is a fraction of moles i.e. the ratio of same quantity, it is unitless. Other concentration terms can be calculated using mole fraction such as molarity, molality, etc.
Complete step-by-step solution:
Mole fraction is the ratio of moles of one substance in a solution mixture to the total number of moles of all substances in the same mixture. For a mixture of two substances, A and B, we can write the mole fractions of each as follows:
\[mole{\text{ }}fraction{\text{ }}of{\text{ }}A = \dfrac{{{n_A}}}{{{n_A} + {n_B}}}\]
We can calculate number of moles of each substance using the below formula:
\[moles = \dfrac{{given{\text{ }}mass}}{{molar{\text{ }}mass}}\]
Number of moles of water are
\[moles = \dfrac{{100}}{{18}} = 5.55moles\]
We are provided that the mole fraction of ethyl alcohol is 0.2 and molecular weight of ethyl alcohol is 46 grams, putting these values in mole fraction formula, we get
\[0.2 = \dfrac{{{m_x}/46}}{{{m_x}/46 + 5.55}}\]
Solving this equation, we get the value of \[{m_x}\]= 64.4 grams.
Therefore, 64.4 grams of ethyl alcohol must be mixed with 100ml of water for the mole fraction (X) of ethyl alcohol to be 0.2.
Hence, the correct option is (D).
Note: The sum of mole fraction of all the components in the solution mixture is always equal to unity. Since it is a fraction of moles i.e. the ratio of same quantity, it is unitless. Other concentration terms can be calculated using mole fraction such as molarity, molality, etc.
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