A 30.5 g sample of an alloy at 93.7°C is placed into 50.1 g water at 23.4°C in an insulated coffee cup. The heat capacity of the coffee cup (without the water) is 9.2 J/K. If the final temperature of the system is 31.1°C, what is the specific heat capacity of the alloy? (c of water is 4.184 J/g×K)

To find the specific heat capacity of the alloy, we need to use the principle of conservation of energy. The heat lost by the alloy will be equal to the heat gained by the water and the coffee cup.

First, let's break down the heat lost by the alloy. The formula to calculate heat lost or gained by a substance is:

Q = mcΔT

Where:
Q is the heat in joules (J)
m is the mass of the substance in grams (g)
c is the specific heat capacity of the substance in J/g×K
ΔT is the change in temperature in degrees Celsius (°C)

For the alloy, the heat lost can be calculated as follows:

Q_alloy = m_alloy × c_alloy × ΔT_alloy

Where:
m_alloy = 30.5 g (mass of the alloy)
c_alloy = specific heat capacity of the alloy (to be determined)
ΔT_alloy = (final temperature - initial temperature) = (31.1°C - 93.7°C)

Next, let's calculate the heat gained by the water and the coffee cup. The total mass of the system is the sum of the mass of water and the mass of the alloy:

m_total = m_water + m_alloy

m_water = 50.1 g (mass of water)

Now, we can calculate the heat gained by the water and the coffee cup. The formula is the same:

Q_water+cup = m_total × c_water+cup × ΔT_total

Where:
Q_water+cup = heat gained by the water and the coffee cup
c_water+cup = specific heat capacity of water and the coffee cup (9.2 J/K)
ΔT_total = (final temperature - initial temperature) = (31.1°C - 23.4°C)

According to the principle of conservation of energy, the heat lost by the alloy is equal to the heat gained by the water and the coffee cup:

Q_alloy = Q_water+cup

Now, we can set up the equation and solve for c_alloy:

m_alloy × c_alloy × ΔT_alloy = m_total × c_water+cup × ΔT_total

30.5 g × c_alloy × (31.1°C - 93.7°C) = (50.1 g + 30.5 g) × 9.2 J/K × (31.1°C - 23.4°C)

Simplifying the equation, we can calculate c_alloy by rearranging the terms:

c_alloy = [(50.1 g + 30.5 g) × 9.2 J/K × (31.1°C - 23.4°C)] / [30.5 g × (31.1°C - 93.7°C)]

Now, substitute the values into the equation and solve for c_alloy to find the specific heat capacity of the alloy.