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A metal M of molar mass 96 reacts with f...

A metal M of molar mass 96 reacts with fluorine to form a salt that can be represeneted as `MF_(x)`. In order to determiner x a 9.18 a sample of the salt is dissolved in 100g of water and its `K_(b)`(water `0.512` kg/mol. Assuming complete dissociation of salf.

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To solve the problem step by step, we will follow the process of determining the value of \( x \) in the salt \( MF_x \) formed by the metal \( M \) and fluorine. ### Step 1: Identify Given Data - Molar mass of metal \( M = 96 \, \text{g/mol} \) - Mass of the salt \( = 9.18 \, \text{g} \) - Mass of water (solvent) \( = 100 \, \text{g} \) - Boiling point elevation constant \( K_b = 0.512 \, \text{°C kg/mol} \) - Boiling point of the solution \( T_b = 374.38 \, \text{K} \) ### Step 2: Calculate the Change in Boiling Point (\( \Delta T_b \)) The change in boiling point is given by: \[ \Delta T_b = T_b - T_{b, \text{pure water}} \] Where \( T_{b, \text{pure water}} = 373 \, \text{K} \) (100 °C). \[ \Delta T_b = 374.38 \, \text{K} - 373 \, \text{K} = 1.38 \, \text{K} \] ### Step 3: Use the Boiling Point Elevation Formula The formula for boiling point elevation is: \[ \Delta T_b = i \cdot K_b \cdot m \] Where: - \( i \) is the van 't Hoff factor (which is 1 for complete dissociation), - \( K_b \) is the ebullioscopic constant, - \( m \) is the molality of the solution. ### Step 4: Rearrange the Formula to Find Molality (\( m \)) Substituting the known values: \[ 1.38 = 1 \cdot 0.512 \cdot m \] \[ m = \frac{1.38}{0.512} \approx 2.695 \, \text{mol/kg} \] ### Step 5: Calculate the Number of Moles of Salt Molality (\( m \)) is defined as: \[ m = \frac{\text{number of moles of solute}}{\text{mass of solvent in kg}} \] The mass of the solvent is \( 100 \, \text{g} = 0.1 \, \text{kg} \). Thus, \[ 2.695 = \frac{n}{0.1} \] \[ n = 2.695 \times 0.1 = 0.2695 \, \text{mol} \] ### Step 6: Relate Moles of Salt to Molar Mass Let the molar mass of the salt \( MF_x \) be \( M_s = 96 + 18x \). The number of moles of the salt is given by: \[ n = \frac{\text{mass of salt}}{\text{molar mass of salt}} = \frac{9.18}{96 + 18x} \] Setting this equal to the number of moles calculated: \[ 0.2695 = \frac{9.18}{96 + 18x} \] ### Step 7: Solve for \( x \) Cross-multiplying gives: \[ 0.2695(96 + 18x) = 9.18 \] Expanding: \[ 25.8912 + 4.851x = 9.18 \] Rearranging: \[ 4.851x = 9.18 - 25.8912 \] \[ 4.851x = -16.7112 \] \[ x \approx 4 \] ### Conclusion Thus, the value of \( x \) in the salt \( MF_x \) is \( 4 \), and the salt can be represented as \( MF_4 \).
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