Question

(a) What is the specific heat of liquid water? (b) What is the molar heat capacity of liquid water? (c) What is the heat capacity of \(185 \mathrm{~g}\) of liquid water? (d) How many \(\mathrm{k}\) ] of heat are needed to raise the temperature of \(10.00 \mathrm{~kg}\) of liquid water from \(24.6^{\circ} \mathrm{C}\) to \(46.2^{\circ} \mathrm{C} ?\)

Short answer

(a) The specific heat capacity of liquid water is approximately \(4.18 J/(g\cdot{^\circ}C)\). (b) The molar heat capacity of liquid water is approximately \(75.33 J/(mol\cdot{^\circ}C)\). (c) The heat capacity of 185 g of liquid water is approximately \(773.3 J/{^\circ}C\). (d) The heat energy required to raise the temperature of 10 kg of water from 24.6 °C to 46.2 °C is approximately \(902.88 \mathrm{kJ}\).

Step-by-step solution

(a) Specific heat capacity of liquid water

The specific heat capacity of liquid water is a constant value, which is approximately \(4.18 J/(g\cdot{^\circ}C)\).

(b) Molar heat capacity of liquid water

To calculate the molar heat capacity of liquid water, we first need to know the molar mass of water. The molar mass of water is \(H_2O = 2 (1.01 g/mol) + 16.00 g/mol ≈ 18.02 g/mol\). Now, we can use the specific heat capacity value and multiply it by the molar mass of water to find the molar heat capacity: Molar heat capacity = Specific heat capacity × Molar mass = \(4.18 J/(g\cdot{^\circ}C) × 18.02 g/mol ≈ 75.33 J/(mol\cdot{^\circ}C)\).

(c) Heat capacity of 185 g of liquid water

To find the heat capacity of 185 g of liquid water, multiply the mass by its specific heat capacity: Heat capacity = Specific heat capacity × Mass = \(4.18 J/(g\cdot{^\circ}C) × 185 g ≈ 773.3 J/{^\circ}C\).

(d) Heat required to raise the temperature

We need to find the heat required to raise the temperature of 10 kg of water from 24.6 °C to 46.2 °C. First, convert the mass to grams: 10 kg = 10,000 g. Next, find the temperature difference: \(ΔT = T_{final} - T_{initial} = 46.2 {^\circ}C - 24.6 {^\circ}C = 21.6 {^\circ}C\). Now, use the formula to calculate the heat energy: Heat energy = Mass × Specific heat capacity × Temperature difference Heat energy = 10,000 g × \(4.18 J/(g\cdot{^\circ}C)\) × 21.6 \(^\circ C ≈ 902,880 J\). Since 1 kJ = 1,000 J, the heat required is approximately \(902.88 \mathrm{kJ}\).

Key concepts

Specific Heat Capacity

Understanding the specific heat capacity is crucial for many scientific calculations, as it determines how much energy is required to raise the temperature of a substance. In the case of liquid water, this value is famously high—a characteristic that contributes to water's effectiveness in storing and transferring heat. The specific heat capacity of water is approximately (4.18 Joules per gram per degree Celsius) ( \(4.18 \ J/(g\cdot^\circ C)\) ). This means that to increase the temperature of 1 gram of water by 1 degree Celsius, it takes 4.18 Joules of energy. Water's high specific heat capacity is why it's such a great coolant and why it takes a while for oceans and lakes to heat up or cool down.

Molar Heat Capacity

Molar heat capacity is another essential concept, especially when dealing with chemical reactions and processes. It describes the amount of heat needed to raise the temperature of one mole of a substance by one degree Celsius. To find the molar heat capacity of water, we combine its specific heat capacity with its molar mass (approximately \(18.02 \ g/mol\) ). The calculation yields a molar heat capacity for water of about 75.33 Joules per mole per degree Celsius ( \(75.33 \ J/(mol\cdot^\circ C)\) ). Understanding this concept helps in various chemical calculations, such as determining the energy changes during reactions.

Thermal Energy Calculation

When we need to calculate the amount of energy required to change the temperature of a substance, we use the formula for thermal energy. Specifically, for water, the formula becomes: Thermal Energy = Mass of water × Specific heat capacity × Temperature change ( \(Q = m \cdot c \cdot \Delta T\) ), where (Q) is the thermal energy in Joules, (m) is the mass in grams, (c) is the specific heat capacity, and (ΔT) is the change in temperature in degrees Celsius. For example, to heat 185 grams of water, which has a specific heat capacity of \(4.18 \ J/(g\cdot^\circ C)\), the amount of energy needed can be found by plugging in the values into the formula. Simple concepts like these underpin more complex thermodynamics.

Temperature Change in Chemistry

Temperature change plays a significant role in chemistry, influencing reaction rates and phase changes. To calculate the heat required to change the temperature of water in a chemistry context, we need to know the initial and final temperatures, the mass of the water, and its specific heat capacity. In our problem, we calculated that to increase the temperature of (10.00 kg) of water from ( 24.6^ ° C) to (46.2^ ° C), we require approximately ( 902.88 kJ) of energy. This practice is fundamental in laboratory settings and industrial processes where precise temperature control is necessary to achieve desired chemical reactions.