The amount of energy released by burning a fuel source, measured in energy per mass, is called the fuel value. If all the energy obtained from burning 1.28 pounds of methane (fuel value is 11.97 kcal/g) is used to heat 127.0 kg of water at an initial temperature of 21.5 °C, what is the final temperature? Help?!!
To solve this problem, we can use the concept of heat transfer:
1. Calculate the total energy obtained from burning 1.28 pounds of methane:
Total energy = Mass of methane * Fuel value
= 1.28 pounds * (11.97 kcal/g) * (453.6 g/pound) [Note: 1 pound = 453.6 grams]
2. Convert the mass of water from kilograms to grams:
Mass of water = 127.0 kg * 1000 g/kg
3. Calculate the heat absorbed by water:
Heat absorbed = Mass of water * Specific heat of water * Change in temperature
Since we want to find the final temperature, we can rearrange the equation as:
Change in temperature = Heat absorbed / (Mass of water * Specific heat of water)
4. Calculate the final temperature:
Final temperature = Initial temperature + Change in temperature
Let's calculate the values step by step:
Step 1:
Mass of methane = 1.28 pounds = 1.28 * 453.6 g = 581.248 g
Total energy = 581.248 g * 11.97 kcal/g
Step 2:
Mass of water = 127.0 kg * 1000 g/kg
Step 3:
To calculate the heat absorbed, we need to know the specific heat of water. The specific heat of water is approximately 4.18 J/g°C.
Heat absorbed = Mass of water * Specific heat of water * Change in temperature
Step 4:
Change in temperature = Heat absorbed / (Mass of water * Specific heat of water)
Final temperature = Initial temperature + Change in temperature
Let's calculate the values to find the final temperature:
To find the final temperature, we can use the concept of specific heat capacity and the equation:
q = mcΔT
Where:
- q is the amount of heat energy transferred
- m is the mass of the substance
- c is the specific heat capacity of the substance
- ΔT is the change in temperature
First, let's calculate the amount of heat energy released by burning 1.28 pounds of methane.
1.28 pounds is approximately 0.581 kilograms (since 1 pound is equal to 0.4536 kilograms).
Next, convert the fuel value from kcal/g to kcal/kg:
11.97 kcal/g = 11.97 kcal/g * 1000 g/kg = 11970 kcal/kg
To find the total amount of heat energy released, multiply the mass of methane burned by its fuel value:
q_methane = 0.581 kg * 11970 kcal/kg = 6946.57 kcal
Now, we need to calculate the amount of heat energy required to heat 127.0 kg of water from 21.5 °C to the final temperature:
q_water = mcΔT
The specific heat capacity of water (c) is approximately 4.186 J/g·°C or 4.186 kcal/kg·°C.
Converting the specific heat capacity to kcal/kg·°C:
4.186 kcal/kg·°C
ΔT can be calculated with the equation:
ΔT = q_water / (mc)
Rearranging the equation to solve for ΔT:
ΔT = q_water / (mc)
Substituting the known values:
ΔT = (6946.57 kcal) / (127.0 kg * 4.186 kcal/kg·°C)
Calculating ΔT:
ΔT ≈ 12.971 °C
Now, to find the final temperature (T_f), we add ΔT to the initial temperature (T_i):
T_f = T_i + ΔT
T_f = 21.5 °C + 12.971 °C
T_f ≈ 34.471 °C
Therefore, the final temperature of the water is approximately 34.471 °C.