Question

Calculate the molality and mole fraction of water, respectively, of a solution that is made by dissolving 3.50 g of potassium nitrate in 50.0 g of water. the final volume of the solution is 56.0 ml.

Answer 1

Answer : The molality of the solution is 0.692m. The mole fraction of water is 0.988.

Explanation :

Molality :

The molality of a solution is calculated as,

$molality = \frac{mol (solute)}{kg (solvent)}$

Step 1 : Find moles of Potassium Nitrate KNO₃.

$mol = \frac{Grams KNO_{3}}{Molar Mass}$

Molar mass of KNO₃ is 101.1 g/mol

Mass of KNO₃ is 3.50 g

$mol KNO_{3} = \frac{3.50g}{101.1g/mol} = 0.0346$

Step 2 : Find kg of the solvent

Water is solvent here.

Mass of water = 50.0 g = 0.050 kg

Step 3 : Use molality formula

Let us use these values to find molality of the solution.

$molality = \frac{0.0346mol}{0.050kg} = 0.692 m$

The molality of the solution is 0.692 m

Mole fraction :

The mole fraction of a component of a mixture is calculated as,

$X_{A} = \frac{molA}{Total Moles}$

Step 1 : Find moles of water

Let us find moles of water using the mole formula

Molar mass of water is 18.02 g/mol

$mol H_{2}O = \frac{50.0 g H_{2}O}{18.02 g/mol} = 2.775$

Step 2 : Use mole fraction formula

Mole fraction of H₂O = $X_{H_{2}O} = \frac{mol H_{2}O}{(mol KNO_{3} + mol H_{2}O)}$

$X_{H_{2}O} = \frac{2.775}{(0.0346 + 2.775)} = \frac{0.0346}{2.8096} = 0.988$

Mole fraction of H₂O is 0.988

Answer 2

0.692 moles/kg and 0.988, respectively.

Further explanation

Given:

• A solution that is made by dissolving 3.50 g of potassium nitrate in 50.0 g of water.
• The molar mass of potassium nitrate (KNO₃) is 101.1 g/moles.
• The molar mass of water (H₂O) is 18.02 g/moles.

Question:

The molality (m) and mole fraction of water (X)

The Process:

Part-1: the molality

The formula for molality is $\boxed{ \ Molality \ m = \frac{moles \ of \ solute}{mass \ of \ solvent \ (kg) } \ }$

which can be made easier as,

$\boxed{ \ m = \frac{mass \ of \ solute \ (g)}{molar \ mass \ (g/moles)} \times \frac{1,000}{mass \ of \ solvent \ (g)} \ }.$ This type of formula will be implemented at present.

Let us calculate the molality of the KNO₃ solution.

$\boxed{ \ m = \frac{3.50 \ (g)}{101.1 \ (g/moles)} \times \frac{1,000}{50.0 \ (g)} \ }.$

Therefore we get the molality of potassium nitrate solution, i.e., 0.692 moles/kg.

Part-2: the mole fraction of water

Let us prepare moles of water solvent and KNO₃ solute.

$\boxed{ \ moles \ of \ water = \frac{mass \ of \ water \ (g)}{molar \ mas\ of \ water \ (g/moles)} \ }$

$\boxed{ \ moles \ of \ water = \frac{50.0 \ g}{18.02 \ g/moles} \ } \rightarrow \boxed{ \ 2.775 \ moles \ of \ water \ }$

$\boxed{ \ moles \ of \ KNO_3 = \frac{mass \ of \ KNO_3 \ (g)}{molar \ mas\ of \ KNO_3 \ (g/moles)} \ }$

$\boxed{ \ moles \ of \ KNO_3 = \frac{3.50 \ g}{101.1 \ g/moles} \ } \rightarrow \boxed{ \ 0.0346 \ moles \ of \ KNO_3 \ }$

The formula for mole fraction of water is $\boxed{ \ X_{water} = \frac{moles \ of \ water}{moles \ of \ water + moles \ of \ KNO_3} \ }$

Let us calculate the mole fraction of water of the KNO₃ solution.

$\boxed{ \ X_{water} = \frac{2.775}{2.775 + 0.0346} \ }$

Thus, the mole fraction of water in KNO₃ solution is 0.988.

Notes:

• The solution comprises a homogeneous mixture of solutes in the solvent.
• In the discussion of solutions, if the solvent is unmentioned, it means the solvent represents water. Water is the most commonly used solvent.
• Concentration states the amount of solute in a solution.
• Concentration represents the intensive property of a solution.

Molality and mole fraction represent several ways of expressing concentration. Furthermore, molality and mole fraction are used in the colligative properties.

• Molality (m) represents the number of moles of solute in every 1 kg (1,000 g) of the solvent.
• Based on suggestions from the National Institute of Standards and Technology in the United States, we accustom it to managing units of molality in moles/kg rather than molal (m). In this manner, we, however, still refer to the SI system of units.
• The mole fraction (X) represents the number of moles of a particular substance compared to the total mole. It is a ratio and a unitless expression.
• The mole fraction of all components of a solution, when added together, will produce one.

$\boxed{ \ X_A = \frac{moles \ of \ A}{moles \ of \ A + moles \ of \ B} \ }$

$\boxed{ \ X_B = \frac{moles \ of \ B}{moles \ of \ A + moles \ of \ B} \ }$

$\boxed{ \ X_A + X_B = 1 \ }$

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Keywords: calculate, the molality, mole fraction of water, a solution, dissolving, potassium nitrate, KNO₃, water, H₂O, the colligative properties