To determine the volume of hydrogen gas generated, we need to follow a series of steps:
### Step 1: Determine the moles of hydrogen gas produced
From the given balanced chemical equation:
[tex]\[ C(s) + H_2O(g) \rightarrow CO(g) + H_2(g) \][/tex]
It is evident that 1 mole of carbon (C) reacts to produce 1 mole of hydrogen gas (H_2). Given that we start with 1.07 mol of carbon, we use the stoichiometric ratio from the equation to find the moles of hydrogen gas produced:
[tex]\[ \text{moles of } H_2 = \text{moles of } C = 1.07 \, \text{mol} \][/tex]
### Step 2: Use the Ideal Gas Law to determine the volume of hydrogen gas
The Ideal Gas Law is expressed as:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure (in atm),
- [tex]\( V \)[/tex] is the volume (in liters),
- [tex]\( n \)[/tex] is the number of moles of gas,
- [tex]\( R \)[/tex] is the gas constant ([tex]\( 0.0821 \, \text{L·atm/(K·mol)} \)[/tex]),
- [tex]\( T \)[/tex] is the temperature (in Kelvin).
We need to solve for the volume [tex]\( V \)[/tex]. Rearrange the Ideal Gas Law to isolate [tex]\( V \)[/tex]:
[tex]\[ V = \frac{nRT}{P} \][/tex]
### Step 3: Plug in the known values
- [tex]\( n = 1.07 \, \text{mol} \)[/tex]
- [tex]\( R = 0.0821 \, \text{L·atm/(K·mol)} \)[/tex]
- [tex]\( T = 315 \, \text{K} \)[/tex]
- [tex]\( P = 1.0 \, \text{atm} \)[/tex]
Substitute these values into the equation:
[tex]\[ V = \frac{1.07 \, \text{mol} \times 0.0821 \, \text{L·atm/(K·mol)} \times 315 \, \text{K}}{1.0 \, \text{atm}} \][/tex]
### Step 4: Calculate the volume
[tex]\[ V = \frac{1.07 \times 0.0821 \times 315}{1} \][/tex]
[tex]\[ V = 27.671805 \, \text{L} \][/tex]
Thus, the volume of hydrogen gas formed from the complete reaction of 1.07 mol of carbon at a pressure of 1.0 atm and a temperature of 315 K is approximately:
[tex]\[ 27.67 \, \text{liters} \][/tex]
### Summary:
- Moles of Hydrogen Gas Produced: [tex]\( 1.07 \, \text{mol} \)[/tex]
- Volume of Hydrogen Gas: [tex]\( 27.67 \, \text{liters} \)[/tex]
This concludes our solution to the problem.
