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Sagot :
To calculate the theoretical yield of carbonic acid ( [tex]\( H_2CO_3 \)[/tex] ) formed when [tex]\( 495 \)[/tex] milliliters of carbon dioxide ([tex]\( CO_2 \)[/tex]) reacts with excess water under the given conditions, we can use the Ideal Gas Law, [tex]\( PV = nRT \)[/tex]. Let's walk through the necessary steps:
1. Convert the Volume to Liters:
Given volume of [tex]\( CO_2 \)[/tex] is [tex]\( 495 \)[/tex] milliliters.
[tex]\[ 495 \text{ milliliters} = 495 \times \frac{1 \text{ liter}}{1000 \text{ milliliters}} = 0.495 \text{ liters} \][/tex]
2. Convert the Temperature to Kelvin:
Given temperature is [tex]\( 25^{\circ} C \)[/tex].
[tex]\[ 25^{\circ} C + 273.15 = 298.15 \text{ K} \][/tex]
3. Use the Ideal Gas Law to Find the Number of Moles ([tex]\( n \)[/tex]) of [tex]\( CO_2 \)[/tex]:
Given pressure [tex]\( P = 101.3 \)[/tex] kilopascals,
Ideal Gas constant [tex]\( R = 8.314 \frac{L kPa}{mol K} \)[/tex],
Temperature [tex]\( T = 298.15 \text{ K} \)[/tex],
Volume [tex]\( V = 0.495 \text{ liters} \)[/tex].
Rearranging the Ideal Gas Law to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Substitute the given values:
[tex]\[ n = \frac{101.3 \text{ kPa} \times 0.495 \text{ liters}}{8.314 \frac{L kPa}{mol K} \times 298.15 \text{ K}} \][/tex]
[tex]\[ n \approx 0.02023 \text{ moles} \][/tex]
4. Calculate the Theoretical Yield of [tex]\( H_2CO_3 \)[/tex]:
According to the balanced chemical equation:
[tex]\[ CO_2 + H_2O \rightarrow H_2CO_3 \][/tex]
1 mole of [tex]\( CO_2 \)[/tex] produces 1 mole of [tex]\( H_2CO_3 \)[/tex]. Thus, moles of [tex]\( H_2CO_3 \)[/tex] formed will be the same as moles of [tex]\( CO_2 \)[/tex], which is [tex]\( 0.02023 \text{ moles} \)[/tex].
5. Calculate the Mass of [tex]\( H_2CO_3 \)[/tex]:
Molar mass of [tex]\( H_2CO_3 \)[/tex] is calculated as follows:
[tex]\[ Molar \, mass \, of \, H_2CO_3 = 2 \times 1 (H) + 12 (C) + 3 \times 16 (O) = 62 \, g/mol \][/tex]
Mass of [tex]\( H_2CO_3 \)[/tex] formed:
[tex]\[ \text{Mass} = \text{moles} \times \text{molar mass} \][/tex]
[tex]\[ \text{Mass} = 0.02023 \text{ moles} \times 62 \, \text{g/mol} \approx 1.254 \, \text{g} \][/tex]
To match the multiple-choice answers:
Option A: [tex]\( 0.889 \, g \)[/tex]
Option B: [tex]\( 1.10 \, g \)[/tex]
Option C: [tex]\( 1.27 \, g \)[/tex]
Option D: [tex]\( 2.029 \, g \)[/tex]
Since [tex]\( 1.254 \)[/tex] grams is closest to [tex]\( 1.27 \)[/tex], the correct answer is:
[tex]\[ \boxed{1.27 \, g} \][/tex]
1. Convert the Volume to Liters:
Given volume of [tex]\( CO_2 \)[/tex] is [tex]\( 495 \)[/tex] milliliters.
[tex]\[ 495 \text{ milliliters} = 495 \times \frac{1 \text{ liter}}{1000 \text{ milliliters}} = 0.495 \text{ liters} \][/tex]
2. Convert the Temperature to Kelvin:
Given temperature is [tex]\( 25^{\circ} C \)[/tex].
[tex]\[ 25^{\circ} C + 273.15 = 298.15 \text{ K} \][/tex]
3. Use the Ideal Gas Law to Find the Number of Moles ([tex]\( n \)[/tex]) of [tex]\( CO_2 \)[/tex]:
Given pressure [tex]\( P = 101.3 \)[/tex] kilopascals,
Ideal Gas constant [tex]\( R = 8.314 \frac{L kPa}{mol K} \)[/tex],
Temperature [tex]\( T = 298.15 \text{ K} \)[/tex],
Volume [tex]\( V = 0.495 \text{ liters} \)[/tex].
Rearranging the Ideal Gas Law to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{PV}{RT} \][/tex]
Substitute the given values:
[tex]\[ n = \frac{101.3 \text{ kPa} \times 0.495 \text{ liters}}{8.314 \frac{L kPa}{mol K} \times 298.15 \text{ K}} \][/tex]
[tex]\[ n \approx 0.02023 \text{ moles} \][/tex]
4. Calculate the Theoretical Yield of [tex]\( H_2CO_3 \)[/tex]:
According to the balanced chemical equation:
[tex]\[ CO_2 + H_2O \rightarrow H_2CO_3 \][/tex]
1 mole of [tex]\( CO_2 \)[/tex] produces 1 mole of [tex]\( H_2CO_3 \)[/tex]. Thus, moles of [tex]\( H_2CO_3 \)[/tex] formed will be the same as moles of [tex]\( CO_2 \)[/tex], which is [tex]\( 0.02023 \text{ moles} \)[/tex].
5. Calculate the Mass of [tex]\( H_2CO_3 \)[/tex]:
Molar mass of [tex]\( H_2CO_3 \)[/tex] is calculated as follows:
[tex]\[ Molar \, mass \, of \, H_2CO_3 = 2 \times 1 (H) + 12 (C) + 3 \times 16 (O) = 62 \, g/mol \][/tex]
Mass of [tex]\( H_2CO_3 \)[/tex] formed:
[tex]\[ \text{Mass} = \text{moles} \times \text{molar mass} \][/tex]
[tex]\[ \text{Mass} = 0.02023 \text{ moles} \times 62 \, \text{g/mol} \approx 1.254 \, \text{g} \][/tex]
To match the multiple-choice answers:
Option A: [tex]\( 0.889 \, g \)[/tex]
Option B: [tex]\( 1.10 \, g \)[/tex]
Option C: [tex]\( 1.27 \, g \)[/tex]
Option D: [tex]\( 2.029 \, g \)[/tex]
Since [tex]\( 1.254 \)[/tex] grams is closest to [tex]\( 1.27 \)[/tex], the correct answer is:
[tex]\[ \boxed{1.27 \, g} \][/tex]
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