Let's solve this problem step-by-step.
### Given Data
1. Cup Heat Capacity (Cp): 46 J/K
2. Volume of Water: 1795 ml
3. Density of Water: 1 g/ml (approximate for water)
4. Mass of Nickel Block: 1000 grams
5. Initial Temperature: 25°C
6. Final Temperature: 125°C
7. Specific Heat of Water (Cp): 4.18 J/g·K
8. Specific Heat of Nickel (Cp): 0.444 J/g·K
9. Specific Heat of Hydrogen (Cp): 142 kJ/g or 142000 J/g
### Step-by-Step Solution
#### Step 1: Convert Water Volume to Mass
Since the density of water is approximately 1 g/ml, the mass of water is:
[tex]\[ \text{Mass of water} = \text{Volume of water} \times \text{Density of water} \][/tex]
[tex]\[ \text{Mass of water} = 1795 \, \text{ml} \times 1 \, \text{g/ml} \][/tex]
[tex]\[ \text{Mass of water} = 1795 \, \text{g} \][/tex]
#### Step 2: Calculate Temperature Change (ΔT)
The temperature change is the final temperature minus the initial temperature:
[tex]\[ \Delta T = 125°C - 25°C \][/tex]
[tex]\[ \Delta T = 100°C \][/tex]
#### Step 3: Calculate Heat Required to Raise the Temperature of Water
The heat ([tex]\(q\)[/tex]) required to raise the temperature of water is:
[tex]\[ q_{water} = \text{mass of water} \times \text{specific heat of water} \times \Delta T \][/tex]
[tex]\[ q_{water} = 1795 \, \text{g} \times 4.18 \, \text{J/g·K} \times 100 \, \text{K} \][/tex]
[tex]\[ q_{water} = 750310 \, \text{J} \][/tex]
#### Step 4: Calculate Heat Required to Raise the Temperature of Nickel
The heat ([tex]\(q\)[/tex]) required to raise the temperature of nickel is:
[tex]\[ q_{nickel} = \text{mass of nickel} \times \text{specific heat of nickel} \times \Delta T \][/tex]
[tex]\[ q_{nickel} = 1000 \, \text{g} \times 0.444 \, \text{J/g·K} \times 100 \, \text{K} \][/tex]
[tex]\[ q_{nickel} = 44400 \, \text{J} \][/tex]
#### Step 5: Calculate Heat Required to Raise the Temperature of the Cup
The heat ([tex]\(q\)[/tex]) required to raise the temperature of the cup is:
[tex]\[ q_{cup} = \text{heat capacity of cup} \times \Delta T \][/tex]
[tex]\[ q_{cup} = 46 \, \text{J/K} \times 100 \, \text{K} \][/tex]
[tex]\[ q_{cup} = 4600 \, \text{J} \][/tex]
#### Step 6: Calculate Total Heat Required
The total heat required to raise the temperature of the system (water, nickel, and cup) is the sum of the heats for each component:
[tex]\[ q_{total} = q_{water} + q_{nickel} + q_{cup} \][/tex]
[tex]\[ q_{total} = 750310 \, \text{J} + 44400 \, \text{J} + 4600 \, \text{J} \][/tex]
[tex]\[ q_{total} = 799310 \, \text{J} \][/tex]
#### Step 7: Calculate the Amount of Hydrogen Gas Required
To find the mass of hydrogen gas required, we use the specific heat of hydrogen:
[tex]\[ \text{mass of hydrogen} = \frac{q_{total}}{\text{specific heat of hydrogen per gram}} \][/tex]
[tex]\[ \text{mass of hydrogen} = \frac{799310 \, \text{J}}{142000 \, \text{J/g}} \][/tex]
[tex]\[ \text{mass of hydrogen} \approx 5.63 \, \text{g} \][/tex]
### Final Answer
You would have to burn approximately 5.63 grams of hydrogen gas to make the final temperature of the system 125°C.
