Get detailed and reliable answers to your questions with IDNLearn.com. Get step-by-step guidance for all your technical questions from our dedicated community members.
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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
