What is the final temperature when 3.0 kg gold bar at 99 Celsius is dropped into 0.22 kg of water at 25 Celsius? (Cp water= 4.186 x10^3 J/kg-C) (Cp gold = 1.29 x10^2 J/kg-C)
To find the final temperature when the gold bar is dropped into water, we can use the principle of energy conservation. The energy lost by the gold bar will be equal to the energy gained by the water.
Let's denote the initial temperature of the gold bar as Tg1, the final temperature of the system as Tf, and the specific heat capacity of gold as Cp_gold. Similarly, let's denote the initial temperature of the water as Tw1, the final temperature of the system as Tf, and the specific heat capacity of water as Cp_water.
Using the principle of energy conservation,
Energy lost by gold bar = Energy gained by water
The energy lost by the gold bar can be calculated as:
Energy lost by gold bar = mass_gold * Cp_gold * (Tg1 - Tf)
The energy gained by the water can be calculated as:
Energy gained by water = mass_water * Cp_water * (Tf - Tw1)
Since energy is conserved, we can equate these two values and solve for Tf.
mass_gold * Cp_gold * (Tg1 - Tf) = mass_water * Cp_water * (Tf - Tw1)
Let's substitute the given values into the equation:
mass_gold = 3.0 kg
Cp_gold = 1.29 x 10^2 J/kg-C
Tg1 = 99°C
mass_water = 0.22 kg
Cp_water = 4.186 x 10^3 J/kg-C
Tw1 = 25°C
Substituting these values into the equation, we get:
3.0 * 1.29 x 10^2 * (99 - Tf) = 0.22 * 4.186 x 10^3 * (Tf - 25)
Simplifying the equation:
387.9 * (99 - Tf) = 925.04 * (Tf - 25)
Now, let's solve for Tf:
387.9 * 99 - 387.9 * Tf = 925.04 * Tf - 925.04 * 25
387.9 * 99 + 925.04 * 25 = 925.04 * Tf + 387.9 * Tf
38343.21 + 23126 = 1312.94 * Tf
Let's find Tf by solving the equation:
TF = (38343.21 + 23126) / 1312.94
TF ≈ 52.48°C
Therefore, the final temperature when the gold bar is dropped into the water is approximately 52.48°C.
To find the final temperature when the gold bar is dropped into the water, we can use the principle of conservation of energy.
The energy lost by the gold bar as it cools down will be gained by the water as it heats up. We can calculate the amount of energy transferred using the specific heat capacity and the change in temperature.
First, let's calculate the energy lost by the gold bar. We can use the formula:
Q_gold = m_gold * Cp_gold * ΔT_gold
Where:
Q_gold is the energy lost by the gold bar
m_gold is the mass of the gold bar (3.0 kg)
Cp_gold is the specific heat capacity of gold (1.29 x 10^2 J/kg-C)
ΔT_gold is the change in temperature of the gold bar (final temperature - initial temperature)
Now, let's calculate the energy gained by the water. We can use the formula:
Q_water = m_water * Cp_water * ΔT_water
Where:
Q_water is the energy gained by the water
m_water is the mass of the water (0.22 kg)
Cp_water is the specific heat capacity of water (4.186 x 10^3 J/kg-C)
ΔT_water is the change in temperature of the water (final temperature - initial temperature)
According to the principle of conservation of energy, the energy lost by the gold bar is equal to the energy gained by the water. Thus, we can set up an equation:
Q_gold = Q_water
m_gold * Cp_gold * ΔT_gold = m_water * Cp_water * ΔT_water
Now, let's solve for the final temperature:
(m_gold * Cp_gold * ΔT_gold) / (m_water * Cp_water) = ΔT_water
Since we know the initial temperatures of the gold bar and the water, we can calculate ΔT_gold and ΔT_water:
ΔT_gold = final temperature of gold - initial temperature of gold = final temperature of gold - 99 C
ΔT_water = final temperature of water - initial temperature of water = final temperature of water - 25 C
By substituting these values into the equation above, we can solve for the final temperature of the water.