Substance Specific heat (J/g·°C) Density (g/cm3)
gold 0.1256 19.3
silver 0.2386 10.5
water 4.184 1.00
A silver cube with an edge length of 2.39 cm and a gold cube with an edge length of 2.73 cm are both heated to 86.9 ∘C and placed in 113.0 mL of water at 19.6 ∘C . What is the final temperature of the water when thermal equilibrium is reached?
To find the final temperature of the water when thermal equilibrium is reached, we can use the concept of heat transfer.
First, let's calculate the heat transferred from the silver cube to the water. We can use the formula:
Q = mcΔT
where:
Q is the heat transferred
m is the mass of the silver cube
c is the specific heat of silver
ΔT is the change in temperature
The mass of the silver cube can be calculated using its density:
density = mass / volume
mass = density x volume
mass of silver cube = density of silver x volume of silver cube
Next, we calculate the volume of the silver cube:
volume of silver cube = (edge length)^3
Now we can substitute the values into the heat transfer formula:
Qsilver = (mass of silver cube) x (specific heat of silver) x (final temperature - initial temperature of silver cube)
Similarly, we can calculate the heat transferred from the gold cube to the water using the same formula:
Qgold = (mass of gold cube) x (specific heat of gold) x (final temperature - initial temperature of gold cube)
The total heat transferred to the water is the sum of the heat transferred from both cubes:
Qtotal = Qsilver + Qgold
Finally, we can use the formula for heat transfer to determine the final temperature of the water:
Qtotal = (mass of water) x (specific heat of water) x (final temperature - initial temperature of water)
Now we can rearrange the formula to solve for the final temperature of the water:
(final temperature - initial temperature of water) = Qtotal / [(mass of water) x (specific heat of water)]
Substitute the known values into the formula and solve for the final temperature.
To find the final temperature of the water when thermal equilibrium is reached, we need to consider the heat transfer between the two metal cubes and the water.
Let's start by calculating the heat transfer for each metal cube using the formula:
Q = m * c * ΔT
where Q is the heat transfer in Joules, m is the mass of the substance in grams, c is the specific heat capacity in J/g·°C, and ΔT is the change in temperature in °C.
For the silver cube:
Mass of silver cube = density * volume = 10.5 g/cm3 * (2.39 cm)3 = 147.61 grams
Heat transfer for silver cube = 147.61 g * 0.2386 J/g·°C * (86.9 °C - 19.6 °C) = 2910.45 J
For the gold cube:
Mass of gold cube = density * volume = 19.3 g/cm3 * (2.73 cm)3 = 244.51 grams
Heat transfer for gold cube = 244.51 g * 0.1256 J/g·°C * (86.9 °C - 19.6 °C) = 2074.74 J
Now, to find the final temperature of the water, we can use the formula for heat transfer:
Q = m * c * ΔT
where Q is the heat transfer in Joules, m is the mass of the water in grams, c is the specific heat capacity of water = 4.184 J/g·°C, and ΔT is the change in temperature in °C.
Rearranging the formula, we can solve for the change in temperature:
ΔT = Q / (m * c)
Now let's calculate the change in temperature of the water:
For water:
Mass of water = 113.0 mL * 1.00 g/cm3 = 113.0 grams
Change in temperature of the water = (2910.45 J + 2074.74 J) / (113.0 g * 4.184 J/g·°C) ≈ 18.74 °C
Finally, to find the final temperature of the water, we can add the change in temperature to the initial temperature:
Final temperature of the water = 19.6 °C + 18.74 °C ≈ 38.34 °C
So, the final temperature of the water will be approximately 38.34 °C when thermal equilibrium is reached.
To solve this problem, we can use the principle of heat transfer. The heat lost by the silver cube and gold cube when they are placed in water is equal to the heat gained by the water. We can use the equation:
Q lost by cubes = Q gained by water
The heat lost by the silver cube can be calculated using the formula:
Q lost = (mass of silver cube) x (specific heat of silver) x (change in temperature)
The mass of the silver cube can be calculated using its density and the equation:
mass of silver cube = (density of silver) x (volume of silver cube)
The heat lost by the gold cube can be calculated using the formula:
Q lost = (mass of gold cube) x (specific heat of gold) x (change in temperature)
The mass of the gold cube can be calculated using its density and the equation:
mass of gold cube = (density of gold) x (volume of gold cube)
The change in temperature for both cubes is the final temperature of the cubes minus the initial temperature of the water:
change in temperature = (final temperature of cubes) - (initial temperature of water)
The heat gained by the water can be calculated using its specific heat, mass, and the same change in temperature:
Q gained = (mass of water) x (specific heat of water) x (change in temperature)
Since the heat lost by the cubes is equal to the heat gained by the water, we can set up the equation:
(mass of silver cube) x (specific heat of silver) x (change in temperature) + (mass of gold cube) x (specific heat of gold) x (change in temperature) = (mass of water) x (specific heat of water) x (change in temperature)
We can solve for the change in temperature using this equation:
(change in temperature) = ((mass of silver cube) x (specific heat of silver) + (mass of gold cube) x (specific heat of gold)) / ((mass of water) x (specific heat of water))
Now we can plug in the given values:
(change in temperature) = ((density of silver) x (volume of silver cube) x (specific heat of silver) + (density of gold) x (volume of gold cube) x (specific heat of gold)) / ((density of water) x (volume of water) x (specific heat of water))
Substituting the values from the table:
(change in temperature) = ((10.5 g/cm3) x (2.39 cm)3 x (0.2386 J/g·°C) + (19.3 g/cm3) x (2.73 cm)3 x (0.1256 J/g·°C)) / ((1.00 g/cm3) x (113.0 mL) x (4.184 J/g·°C))
After evaluating this expression, we can add the result to the initial temperature of the water to get the final temperature of the water when thermal equilibrium is reached.