A piece of iron (Cp = 0.450 J/(gi°C)) that has a mass of 21.5 g and an initial temperature of 100.0°C is submerged in X g of water (Cp = 4.184 J/(gi°C)) at 20.0°C. The temperature of the system (iron and water) changes to 21.4°C. What is the value of X?

heat lost by Fe + heat gained by water = 0

[mass Fe x specific heat Fe x (Tfinal-Tinitial)] + [mass H2O x specific heat H2O x (Tfinal-Tinitial)] = 0
Substitute and solve for mass H2O which is the only unknown.

To find the value of X, we need to use the principle of conservation of energy and the formula for heat transfer.

The heat gained by the water is equal to the heat lost by the iron.

The formula for heat transfer is:

Q = m * Cp * ΔT

where:
Q = heat transfer
m = mass
Cp = specific heat capacity
ΔT = change in temperature

For the water: Qwater = (X g) * (4.184 J/(g·°C)) * (21.4°C - 20.0°C)

For the iron: Qiron = (21.5 g) * (0.450 J/(g·°C)) * (100.0°C - 21.4°C)

According to the conservation of energy principle:

Qwater = Qiron

Therefore:

(X g) * (4.184 J/(g·°C)) * (21.4°C - 20.0°C) = (21.5 g) * (0.450 J/(g·°C)) * (100.0°C - 21.4°C)

Now we can solve for X by rearranging the equation:

(X g) = [(21.5 g) * (0.450 J/(g·°C)) * (100.0°C - 21.4°C)] / [(4.184 J/(g·°C)) * (21.4°C - 20.0°C)]

Simplifying the equation:

(X g) = (21.5 g) * (0.450 J/(g·°C)) * (100.0°C - 21.4°C) / [(4.184 J/(g·°C)) * (21.4°C - 20.0°C)]

Now we can calculate X:

X = ((21.5 g) * (0.450 J/(g·°C)) * (100.0°C - 21.4°C)) / ((4.184 J/(g·°C)) * (21.4°C - 20.0°C))

Plugging in the values and calculating X:

X ≈ 248.44 g