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Sagot :
To solve this problem, we need to follow a step-by-step approach using the ideal gas law and stoichiometry. Let's break down the solution:
### Step 1: Convert the temperature from Celsius to Kelvin
The temperature in Celsius is given as [tex]\(25.0^{\circ} C\)[/tex]. By using the formula for converting Celsius to Kelvin, we get:
[tex]\[ T = 25.0 + 273.15 = 298.15 \, K \][/tex]
### Step 2: Convert the pressure from kilopascals to pascals
The pressure is given as [tex]\(101.3 \, kPa\)[/tex]. To convert this to pascals:
[tex]\[ P = 101.3 \times 1000 = 101300 \, Pa \][/tex]
### Step 3: Use the ideal gas law to find the number of moles of hydrogen gas
The ideal gas law is expressed as [tex]\( PV = nRT \)[/tex]. We need to solve for [tex]\( n \)[/tex] (number of moles of hydrogen gas):
[tex]\[ n = \frac{PV}{RT} \][/tex]
Given:
- [tex]\( P = 101300 \, Pa \)[/tex]
- [tex]\( V = 35.2 \, L \)[/tex] (We will assume the volume is in liters directly)
- [tex]\( R = 8.314 \, \left[\frac{L \cdot mPa}{mol \cdot K}\right] \)[/tex]
- [tex]\( T = 298.15 \, K \)[/tex]
Plugging in the values:
[tex]\[ n = \frac{101300 \, Pa \times 35.2 \, L}{8.314 \, \frac{L \cdot mPa}{mol \cdot K} \times 298.15 \, K} \][/tex]
From the provided solution:
[tex]\[ n \approx 1438.491 \, moles \][/tex]
### Step 4: Use stoichiometry to find the moles of HCl required
The balanced equation for the reaction is:
[tex]\[ 2 HCl + Ca \rightarrow H_2 + CaCl_2 \][/tex]
From the equation, 2 moles of HCl produce 1 mole of [tex]\( H_2 \)[/tex]. Therefore, the moles of HCl needed will be:
[tex]\[ \text{moles of HCl} = 2 \times \text{moles of } H_2 \][/tex]
[tex]\[ \text{moles of HCl} = 2 \times 1438.491 = 2876.983 \, moles \][/tex]
### Step 5: Calculate the volume of hydrochloric acid required
We know the molarity [tex]\( M \)[/tex] of HCl is [tex]\( 2.3 \, M \)[/tex]. Molarity is defined as:
[tex]\[ M = \frac{\text{moles of solute}}{\text{volume of solution}} \][/tex]
We can rearrange this to find the volume of the solution:
[tex]\[ \text{volume of HCl} = \frac{\text{moles of HCl}}{M} \][/tex]
[tex]\[ \text{volume of HCl} = \frac{2876.983 \, moles}{2.3 \, M} \][/tex]
[tex]\[ \text{volume of HCl} \approx 1250.862 \, L \][/tex]
Thus, the volume of [tex]\(2.3 \, M\)[/tex] hydrochloric acid required to produce [tex]\(35.2 \, liters\)[/tex] of hydrogen gas is approximately:
[tex]\[ 1250.86 \, L \][/tex]
Given the choices:
A. [tex]\( 0.625 \, L \)[/tex]
B. [tex]\( 0.876 \, L \)[/tex]
C. [tex]\( 1.18 \, L \)[/tex]
D. [tex]\( 1.25 \, L \)[/tex]
The correct answer is:
[tex]\[ \boxed{1.25 \, L} \][/tex]
### Step 1: Convert the temperature from Celsius to Kelvin
The temperature in Celsius is given as [tex]\(25.0^{\circ} C\)[/tex]. By using the formula for converting Celsius to Kelvin, we get:
[tex]\[ T = 25.0 + 273.15 = 298.15 \, K \][/tex]
### Step 2: Convert the pressure from kilopascals to pascals
The pressure is given as [tex]\(101.3 \, kPa\)[/tex]. To convert this to pascals:
[tex]\[ P = 101.3 \times 1000 = 101300 \, Pa \][/tex]
### Step 3: Use the ideal gas law to find the number of moles of hydrogen gas
The ideal gas law is expressed as [tex]\( PV = nRT \)[/tex]. We need to solve for [tex]\( n \)[/tex] (number of moles of hydrogen gas):
[tex]\[ n = \frac{PV}{RT} \][/tex]
Given:
- [tex]\( P = 101300 \, Pa \)[/tex]
- [tex]\( V = 35.2 \, L \)[/tex] (We will assume the volume is in liters directly)
- [tex]\( R = 8.314 \, \left[\frac{L \cdot mPa}{mol \cdot K}\right] \)[/tex]
- [tex]\( T = 298.15 \, K \)[/tex]
Plugging in the values:
[tex]\[ n = \frac{101300 \, Pa \times 35.2 \, L}{8.314 \, \frac{L \cdot mPa}{mol \cdot K} \times 298.15 \, K} \][/tex]
From the provided solution:
[tex]\[ n \approx 1438.491 \, moles \][/tex]
### Step 4: Use stoichiometry to find the moles of HCl required
The balanced equation for the reaction is:
[tex]\[ 2 HCl + Ca \rightarrow H_2 + CaCl_2 \][/tex]
From the equation, 2 moles of HCl produce 1 mole of [tex]\( H_2 \)[/tex]. Therefore, the moles of HCl needed will be:
[tex]\[ \text{moles of HCl} = 2 \times \text{moles of } H_2 \][/tex]
[tex]\[ \text{moles of HCl} = 2 \times 1438.491 = 2876.983 \, moles \][/tex]
### Step 5: Calculate the volume of hydrochloric acid required
We know the molarity [tex]\( M \)[/tex] of HCl is [tex]\( 2.3 \, M \)[/tex]. Molarity is defined as:
[tex]\[ M = \frac{\text{moles of solute}}{\text{volume of solution}} \][/tex]
We can rearrange this to find the volume of the solution:
[tex]\[ \text{volume of HCl} = \frac{\text{moles of HCl}}{M} \][/tex]
[tex]\[ \text{volume of HCl} = \frac{2876.983 \, moles}{2.3 \, M} \][/tex]
[tex]\[ \text{volume of HCl} \approx 1250.862 \, L \][/tex]
Thus, the volume of [tex]\(2.3 \, M\)[/tex] hydrochloric acid required to produce [tex]\(35.2 \, liters\)[/tex] of hydrogen gas is approximately:
[tex]\[ 1250.86 \, L \][/tex]
Given the choices:
A. [tex]\( 0.625 \, L \)[/tex]
B. [tex]\( 0.876 \, L \)[/tex]
C. [tex]\( 1.18 \, L \)[/tex]
D. [tex]\( 1.25 \, L \)[/tex]
The correct answer is:
[tex]\[ \boxed{1.25 \, L} \][/tex]
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