Question

Calculate the amount of energy required (in joules) to heat \(2.5 \mathrm{kg}\) of water from \(18.5^{\circ} \mathrm{C}\) to \(55.0^{\circ} \mathrm{C}\).

Short answer

The amount of energy required to heat \(2.5\:kg\) of water from \(18.5^{\circ} C\) to \(55.0^{\circ} C\) is \(382075\:J\).

Step-by-step solution

Identify the known values

We know the mass of the water (\(m = 2.5\:kg\)), the initial temperature of the water (\(T_i = 18.5^{\circ} C\)), and the final temperature of the water (\(T_f = 55.0^{\circ} C\)). Also, we have the specific heat capacity of water (\(c = 4.18\:J/(g K))\).

Calculate the temperature change

To find the change in temperature, we need to subtract the initial temperature from the final temperature. That is: \(ΔT = T_f - T_i\). Calculate the value of \(ΔT\): \(ΔT = 55.0^{\circ} C - 18.5^{\circ} C = 36.5^{\circ} C\)

Convert mass to grams

The formula for energy calculation requires the mass to be in grams. To convert the mass of water from kilograms to grams, we need to multiply by 1000: \(m = 2.5\:kg \times 1000 = 2500\:g\)

Calculate the amount of energy required

Now we can plug the values we found into the formula for calculating energy, \(q = mcΔT\): \(q = (2500\:g) (4.18\, J/(g K)) (36.5^{\circ} C)\) \(q = 382075\:J\) The amount of energy required to heat the water is \(382075\:J\).

Key concepts

Specific Heat Capacity

Specific heat capacity is a crucial concept when dealing with the heating or cooling of substances. It is defined as the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius (or Kelvin). The specific heat capacity is denoted by the symbol \(c\) and is typically expressed in units of joules per gram per degree Celsius \((J/g°C)\).
The specific heat capacity of water is particularly high at \(4.18 \ J/(g°C)\), which means water can absorb a lot of heat without experiencing a large change in temperature. This unique property makes water an excellent coolant and is why it's often used in heating and cooling applications.

Temperature Change

To determine the energy required for heating a substance, a key figure to know is the temperature change. The temperature change, \(ΔT\), is the difference between the final temperature (\(T_f\)) and the initial temperature (\(T_i\)). It's calculated simply by:

• \(ΔT = T_f - T_i\)

In the provided example, to find how much energy is needed to heat the water, we found a temperature change of \(36.5^{\circ} \, C\). This difference is vital because it directly influences the total energy calculated.

Mass Conversion

Sometimes, the measurements used in scientific calculations require conversions for consistency. Mass conversion is frequently necessary when working with specific heat capacity formulas because these formulas often use grams, rather than kilograms, for mass.
In the energy calculation exercise, it's essential to convert the water's mass from kilograms to grams for accurate calculations, since the specific heat capacity is given in \(g\). You convert the mass by multiplying:

• \(2.5 \, kg \times 1000 = 2500 \, g\)

This conversion ensures the compatibility of units, enabling the correct application of formulas.

Joules

Joules are a unit of energy used in the International System of Units (SI). In thermal calculations, joules quantify the amount of energy transferred as heat. The specific heat capacity, mass, and temperature change all contribute to determining the amount of energy in joules required to heat or cool a substance.
In the context of the exercise, once you have the correct values for mass (in grams), specific heat capacity, and temperature change, you can compute the total energy in joules using the formula:

• \(q = mcΔT\)

For example, with a calculated energy of \(382075 \, J\), it reflects the total energy input needed to achieve the desired temperature rise.