How many joules of energy are needed to melt 15.5 g of ice and heat it to $27.0^{\circ} C$?
Answer (1)
Calculate the energy Q 1 required to melt the ice using Q 1 = m L f = 15.5 g 334 J/g = 5177 J .
Calculate the energy Q 2 required to heat the water from 0 ∘ C to 27. 0 ∘ C using Q 2 = m c Δ T = 15.5 g 4.186 J/g ⋅ ∘ C 27. 0 ∘ C = 1751.841 J .
Calculate the total energy Q t o t a l by summing Q 1 and Q 2 : Q t o t a l = Q 1 + Q 2 = 5177 J + 1751.841 J = 6928.841 J .
The total energy required is 6928.841 J .
Explanation
Problem Analysis We are given the mass of ice, m = 15.5 g, and we want to find the total energy required to melt this ice and heat it to a final temperature of 27. 0 ∘ C . The ice starts at an initial temperature of 0 ∘ C . We will need to use the latent heat of fusion of ice, L f = 334 J/g, and the specific heat capacity of water, c = 4.186 J/g ⋅ ∘ C .
Energy to Melt the Ice First, we calculate the energy required to melt the ice. This is given by the formula: Q 1 = m ⋅ L f
where m is the mass of the ice and L f is the latent heat of fusion. Plugging in the values, we get: Q 1 = 15.5 g ⋅ 334 J/g = 5177 J So, 5177 Joules are required to melt the ice.
Energy to Heat the Water Next, we calculate the energy required to heat the water from 0 ∘ C to 27. 0 ∘ C . This is given by the formula: Q 2 = m ⋅ c ⋅ Δ T
where m is the mass of the water (which is the same as the initial mass of the ice), c is the specific heat capacity of water, and Δ T is the change in temperature. Plugging in the values, we get: Q 2 = 15.5 g ⋅ 4.186 J/g ⋅ ∘ C ⋅ ( 27. 0 ∘ C − 0 ∘ C ) = 15.5 g ⋅ 4.186 J/g ⋅ ∘ C ⋅ 27. 0 ∘ C = 1751.841 J So, 1751.841 Joules are required to heat the water to 27. 0 ∘ C .
Total Energy Required Finally, we calculate the total energy required by summing the energy required for melting and the energy required for heating: Q t o t a l = Q 1 + Q 2 = 5177 J + 1751.841 J = 6928.841 J Therefore, the total energy required is 6928.841 Joules.
Final Answer The total energy needed to melt 15.5 g of ice and heat it to 27. 0 ∘ C is approximately 6928.841 Joules.
Examples
Understanding the energy needed for phase transitions and temperature changes is crucial in many real-world applications. For example, when designing cooling systems for electronics, engineers need to calculate the amount of heat that needs to be removed to keep the components at a safe operating temperature. Similarly, in food processing, knowing the energy required to freeze or thaw food helps optimize the process and maintain food quality. This principle is also applied in climate science to model the melting of glaciers and ice caps and its impact on sea levels, where the energy required to melt ice is a critical factor.
