The specific heat capacity of a pure substance can be found by dividing the heat needed to change the temperature of a sample of the substance by the mass of the sample and by the change in temperature. The heat capacity of a certain substance has been measured to be $1.66 \frac{J}{ g \cdot{ }^{\circ} C }$. Suppose 149 g of the substance are heated until the temperature of the sample has changed by $42.9^{\circ} C$.
Write an equation that will let you calculate the heat $\underline{Q}$ that was needed for this temperature change. Your equation should contain only symbols. Be sure you define each symbol.
Your equation:
$Q=\square$
Definitions of your symbols:
$\square=1.66 \frac{J}{g \cdot{ }^{\circ} C}$
$\square$ - $149 . g$
$\square$ $=42.9^{\circ} C$
Answer (1)
The problem provides the specific heat capacity, mass, and change in temperature of a substance and asks for the heat required for the temperature change.
The formula c = m Δ T Q is rearranged to solve for Q , resulting in Q = c ⋅ m ⋅ Δ T .
The symbols are defined as: Q (heat), c (specific heat capacity), m (mass), and Δ T (change in temperature).
The final equation to calculate the heat is: Q = c ⋅ m ⋅ Δ T .
Explanation
Understanding the Problem The problem states that the specific heat capacity of a substance is found by dividing the heat needed to change the temperature of a sample by the mass of the sample and the change in temperature. We are given the specific heat capacity ( c ), the mass of the substance ( m ), and the change in temperature ( Δ T ). Our goal is to find an equation for the heat ( Q ) needed for this temperature change.
Finding the Equation for Heat The formula for specific heat capacity is: c = m Δ T Q To find the heat Q , we need to rearrange this formula to solve for Q .
The Equation Multiply both sides of the equation by m Δ T :
Q = c m Δ T This equation tells us that the heat required is equal to the specific heat capacity multiplied by the mass and the change in temperature.
Defining the Symbols Now, let's define each symbol:
Q = heat needed for the temperature change (in Joules)
c = specific heat capacity (in g ⋅ ∘ C J )
m = mass of the substance (in grams)
Δ T = change in temperature (in ∘ C )
Calculating the Heat We are given:
c = 1.66 g ⋅ ∘ C J
m = 149 g
Δ T = 42.9 ∘ C
Plugging these values into the equation: Q = 1.66 g ⋅ ∘ C J × 149 g × 42.9 ∘ C Q = 10610.886 J Therefore, the heat needed for this temperature change is approximately 10610.886 Joules.
Examples
Understanding specific heat capacity is crucial in many real-world applications. For instance, when designing engines, engineers need to know how much heat different materials can absorb without undergoing significant temperature changes. This helps in selecting the right materials to prevent overheating and ensure efficient operation. Similarly, in cooking, understanding the specific heat of water helps in determining how much energy is needed to boil water for pasta or to steam vegetables.
