An aluminum calorimetry cup [mc = 0.6 kg, cc = 0.22 kcal/(kgC)] is filled with water [mw = 2.4 kg, cw = 1 kcal/(kgC)]. It is at a temperature of 30C. A sample of an unknown substance with mass mm = 2.34 kg, originally at a temperature of 210C, is added. After a while, the temperature of the water is 35C. What is the specific heat of this material?
name change
To find the specific heat of the unknown substance, we can use the principle of energy conservation. The energy lost by the hot substance (q_loss) is equal to the energy gained by the water (q_gain) and the calorimetry cup.
The formula for energy change is given by:
q = mcΔT
where q is the energy change, m is the mass, c is the specific heat, and ΔT is the temperature change.
First, let's calculate the energy change of the water:
q_gain_water = mw * cw * ΔT_water
where mw is the mass of water, cw is the specific heat of water, and ΔT_water is the change in temperature of the water.
ΔT_water = 35C - 30C = 5C
q_gain_water = 2.4 kg * 1 kcal/(kgC) * 5C = 12 kcal
Next, let's calculate the energy change of the calorimetry cup:
q_gain_cup = mc * cc * ΔT_cup
where mc is the mass of the calorimetry cup, cc is the specific heat of the calorimetry cup, and ΔT_cup is the change in temperature of the calorimetry cup.
ΔT_cup is the same as the ΔT_water because the cup and water are in thermal equilibrium.
q_gain_cup = 0.6 kg * 0.22 kcal/(kgC) * 5C = 0.66 kcal
Now, let's calculate the energy lost by the hot substance:
q_loss_substance = mm * cm * ΔT_substance
where mm is the mass of the unknown substance, cm is the specific heat of the unknown substance, and ΔT_substance is the change in temperature of the unknown substance.
ΔT_substance = 210C - 35C = 175C
We can rearrange the equation to solve for cm:
cm = q_loss_substance / (mm * ΔT_substance)
Finally, let's substitute the values and calculate the specific heat of the unknown substance:
cm = (q_loss_substance) / (mm * ΔT_substance)
Now, to calculate q_loss_substance, we can rearrange the equation for energy change:
q_loss = q_gain_water + q_gain_cup (according to the principle of energy conservation)
q_loss_substance = q_loss - q_gain_water - q_gain_cup
Substituting the given values, we have:
q_loss_substance = q_loss - 12 kcal - 0.66 kcal
Now, let's calculate the value of q_loss:
q_loss = mc * cc * ΔT_substance
q_loss = 0.6 kg * 0.22 kcal/(kgC) * 175C
Substituting the values into the equation, we have:
q_loss_substance = (0.6 kg * 0.22 kcal/(kgC) * 175C) - 12 kcal - 0.66 kcal
After performing the calculations, we can determine the value of q_loss_substance. Dividing q_loss_substance by (mm * ΔT_substance), we can find the specific heat of the unknown substance (cm).