A sample of an unknown substance has a mass of 0.158 kg. If [tex]$2,510.0 J$[/tex] of heat is required to heat the substance from [tex]$32.0^{\circ} C$[/tex] to [tex]$61.0^{\circ} C$[/tex], what is the specific heat of the substance?
Use [tex]$q =m C _p \Delta T$[/tex].
Answer (1)
Calculate the change in temperature: Δ T = 61.0 − 32.0 = 29. 0 ∘ C .
Rearrange the formula to solve for specific heat: C p = m Δ T q .
Substitute the given values: C p = 158 × 29.0 2510.0 .
Calculate the specific heat: C p ≈ 0.548 J / ( g " , ∘ C ) .
Explanation
Problem Analysis and Given Data We are given the mass of a substance, the amount of heat required to raise its temperature, and the initial and final temperatures. We need to find the specific heat of the substance using the formula q = m C p Δ T , where:
q is the heat energy (in Joules)
m is the mass of the substance (in grams)
C p is the specific heat capacity (in J / ( g " , ∘ C ) )
Δ T is the change in temperature (in ∘ C )
We have:
m = 0.158 k g = 158 g
q = 2510.0 J
T i = 32. 0 ∘ C
T f = 61. 0 ∘ C
Calculate the Change in Temperature First, we need to calculate the change in temperature, Δ T :
Δ T = T f − T i = 61. 0 ∘ C − 32. 0 ∘ C = 29. 0 ∘ C
Rearrange the Formula Now, we rearrange the formula q = m C p Δ T to solve for C p :
C p = m Δ T q
Substitute Values and Calculate Specific Heat Next, we substitute the given values into the formula: C p = 158 g × 29. 0 ∘ C 2510.0 J C p = 158 × 29.0 2510.0 ≈ 0.54779572239 J / ( g " , ∘ C ) Rounding to three significant figures, we get: C p ≈ 0.548 J / ( g " , ∘ C )
Final Answer Therefore, the specific heat of the substance is approximately 0.548 J / ( g " , ∘ C ) .
Examples
Understanding specific heat is crucial in many real-world applications. For example, when designing cooling systems for electronics, engineers need to know the specific heat of the materials used to dissipate heat effectively. Similarly, in cooking, different materials have different specific heats, which affects how quickly they heat up or cool down. Knowing the specific heat helps in selecting the right cookware for even cooking.
