[mass Au x specific heat Au x (Tfinal-Tinitial)] + [mass H2O x specific heat water x (Tfinal-Tinitial)] = 0
Solve for Tf.
Solve for Tf.
The heat lost by the gold can be calculated using the equation:
Q_gold = m_gold * c_gold * (T_final - T_initial)
Where:
Q_gold is the heat lost by the gold,
m_gold is the mass of the gold (which is not given),
c_gold is the specific heat capacity of gold,
T_final is the final temperature of the gold, coffee cup, and water, and
T_initial is the initial temperature of the gold.
The heat gained by the water and the coffee cup can be calculated using the equation:
Q_water + Q_cup = m_water * c_water * (T_final - T_initial_water)
Where:
Q_water is the heat gained by the water,
Q_cup is the heat gained by the coffee cup,
m_water is the mass of the water,
c_water is the specific heat capacity of water,
T_final is the final temperature of the gold, coffee cup, and water, and
T_initial_water is the initial temperature of the water.
Since there is no heat exchange between the gold and the coffee cup, we have:
Q_gold = -Q_cup
Substituting the equations, we get:
m_gold * c_gold * (T_final - T_initial) = -[m_water * c_water * (T_final - T_initial_water) + C_cup * (T_final - T_initial)]
Where:
C_cup is the heat capacity of the coffee cup.
Now, let's solve the equation for T_final.
First, let's find the mass of water:
m_water = 50 g
Next, let's find the initial temperature of the water:
T_initial_water = 300 K
Now, let's substitute the values into the equation and solve for T_final:
m_gold * c_gold * (T_final - 350 K) = -[50 g * 4.18 J/g°C * (T_final - 300 K) + 400 J/°C * (T_final - 350 K)]
Simplifying the equation gives us:
m_gold * c_gold * (T_final - 350 K) = -[209 J/(°C g) * (T_final - 300 K) + 400 J/°C * (T_final - 350 K)]
Now, you can solve this equation for T_final by rearranging and simplifying further.
Note: Since the mass of the gold is not given in the question, you would need that information to calculate the final temperature.