A 325.0-g piece of gold at 427°C is dropped into 200.0 g of water at 22.0°C. (The specific heat of gold is 0.131 J/g°C.) Calculate the final temperature of the mixture. (Assume no heat loss to the surroundings.)
Can you please outline step by step how i would solve this problem?
Sure, let's break it down step by step:
1. Calculate the heat lost by the gold:
- Use the formula: Q = m * c * ΔT, where Q is the heat lost, m is the mass of gold, c is the specific heat of gold, and ΔT is the change in temperature.
- Plug in the given values: m = 325.0 g, c = 0.131 J/g°C, ΔT = (final temperature - initial temperature) = (final temperature - 427°C).
- This gives us: Q = 325.0 g * 0.131 J/g°C * (final temperature - 427°C).
2. Calculate the heat gained by the water:
- Use the formula: Q = m * c * ΔT, where Q is the heat gained, m is the mass of water, c is the specific heat of water (4.18 J/g°C, approximately), and ΔT is the change in temperature.
- Plug in the given values: m = 200.0 g, c = 4.18 J/g°C, and ΔT = (final temperature - 22.0°C).
- This gives us: Q = 200.0 g * 4.18 J/g°C * (final temperature - 22.0°C).
3. Since the heat lost by the gold is equal to the heat gained by the water (Q = Q), we can set these two equations equal to each other and solve for the final temperature.
4. Simplify and solve the equation:
- Set the equations equal to each other: 325.0 g * 0.131 J/g°C * (final temperature - 427°C) = 200.0 g * 4.18 J/g°C * (final temperature - 22.0°C).
- Simplify and rearrange the equation to solve for the final temperature.
5. Solve the equation for the final temperature:
- Once you have the equation from step 4, you can solve it to find the final temperature. You can use algebraic manipulation to get the final temperature on one side of the equation and the constants on the other side.
- Remember to check your units and conversions throughout the calculation.
6. Once you have the final temperature, don't forget to include the units and round to an appropriate number of significant figures.
And there you have it, a step-by-step outline to solve the problem. Good luck!
Sure! Here's a step-by-step guide on how to solve the problem:
Step 1: Determine the heat gained or lost by gold and water separately.
- To calculate the heat gained or lost, use the formula Q = m * c * ΔT, where Q is the heat, m is the mass, c is the specific heat, and ΔT is the change in temperature.
- For the gold, substitute the values into the formula: Qgold = mgold * cgold * ΔTgold.
- For the water, substitute the values into the formula: Qwater = mwater * cwater * ΔTwater.
Step 2: Set up an equation to find the final temperature.
- The heat lost by the gold is equal to the heat gained by the water because no heat is lost to the surroundings.
- Therefore, Qgold = -Qwater. The negative sign indicates a heat transfer from gold to water.
- Substitute the values into the equation: mgold * cgold * ΔTgold = -mwater * cwater * ΔTwater.
Step 3: Rearrange the equation to solve for the final temperature.
- Use the equation: ΔTwater / ΔTgold = -mgold * cgold / -mwater * cwater.
- Simplify the equation by canceling out common terms: ΔTwater / ΔTgold = mgold * cgold / mwater * cwater.
- Rearrange the equation to solve for ΔTwater: ΔTwater = (mgold * cgold * ΔTgold) / (mwater * cwater).
- Substitute the known values into the equation.
Step 4: Calculate the final temperature.
- The final temperature (Tfinal) can be found by adding the change in temperature (ΔT) to the initial temperature (Tinitial).
- Use the equation: Tfinal = Tinitial + ΔTwater.
- Substitute the known initial temperature of the water into the equation.
- Calculate the final temperature using the equation and the calculated ΔTwater.
That's it! By following these steps, you can find the final temperature of the mixture.
Sure! Here's a step-by-step guide to help you solve this problem:
Step 1: Determine the heat gained or lost by the gold.
To do this, use the formula Q = mcΔT, where Q represents the heat gained or lost, m represents the mass of the substance, c represents the specific heat capacity, and ΔT represents the change in temperature.
Q(gold) = m(gold) * c(gold) * ΔT(gold)
Substituting in the given values:
m(gold) = 325.0 g
c(gold) = 0.131 J/g°C
ΔT(gold) = final temperature - initial temperature = final temperature - 427°C
Step 2: Determine the heat gained or lost by the water.
Using the same formula as in Step 1, but now for water:
Q(water) = m(water) * c(water) * ΔT(water)
Substituting in the given values:
m(water) = 200.0 g
c(water) = 4.18 J/g°C (specific heat capacity of water)
ΔT(water) = final temperature - initial temperature = final temperature - 22.0°C
Step 3: Apply the principle of conservation of energy.
Since there is no heat loss to the surroundings, the heat gained by the gold will be equal to the heat gained by the water.
Q(gold) = Q(water)
Step 4: Equate the heat equations.
Equate the equations from Step 1 and Step 2:
m(gold) * c(gold) * ΔT(gold) = m(water) * c(water) * ΔT(water)
Step 5: Rearrange the equation to solve for the final temperature.
Rearrange the equation from Step 4 to solve for the final temperature:
final temperature = (m(gold) * c(gold) * ΔT(gold) / (m(water) * c(water))) + initial temperature
Substitute the given values and solve for the final temperature.
I hope this helps! Let me know if you have any further questions.