If you wish to warm 52 kg of water by 22 ∘c for your bath, find what the quantity of heat is needed. express your answer in calories cal.

To find out the quantity of heat needed to warm the water, you can use the formula for heat energy transfer:

\[ Q = m \times c \times \Delta T \]

Where: - \( Q \) is the heat energy in joules (J) - \( m \) is the mass of the water in kilograms (kg) - \( c \) is the specific heat capacity of water, which is \( 4.186 \) joules per gram per degree Celsius (J/g°C) - \( \Delta T \) is the change in temperature in degrees Celsius (°C)

However, since you want the answer in calories (cal), remember that 1 calorie is the energy needed to raise the temperature of 1 gram of water by 1°C. There is also the conversion factor between joules and calories, which is \( 1 \text{ cal} = 4.184 \text{ J} \).

First, convert the mass from kilograms to grams (since the specific heat capacity is in terms of grams):

\( m = 52 \text{ kg} \times 1000 \text{ g/kg} = 52000 \text{ g} \)

Then use the provided data: - \( m = 52000 \text{ g} \) - \( c \) for water is \( 1 \text{ cal/g°C} \) when you want to find the heat in calories - \( \Delta T = 22 °C \)

Calculate the heat energy required:

\[ Q = m \times c \times \Delta T \] \[ Q = 52000 \text{ g} \times 1 \text{ cal/g°C} \times 22 °C \] \[ Q = 1144000 \text{ cal} \]

So, the quantity of heat needed to warm 52 kg of water by 22°C for your bath is 1,144,000 calories.

Extra: The concept of heat transfer is a fundamental aspect of thermodynamics, which is the branch of physics that deals with the relationships between heat and other forms of energy. When you're heating water for a bath, you're transferring energy from your heat source (like a water heater) to the water. The specific heat capacity of a substance is the amount of energy required to raise the temperature of one gram of the substance by one degree Celsius. Water has a high specific heat capacity, which means it can absorb quite a bit of heat before its temperature rises significantly.

In practical terms, this means that heating water for purposes like baths requires a substantial amount of energy. By knowing the mass of the water and the temperature increase desired, you can calculate the energy needed, typically using a heater or boiler. Efficiency of these devices can vary, and not all the energy consumed will be transferred to the water as usable heat due to losses like heat escaping to the surrounding environment.

It's also important to understand the difference between the calorie used in chemistry and physics (the amount of energy needed to raise the temperature of 1 gram of water by 1°C) and the Calorie (with a capital "C") used in food (which is equivalent to 1000 chemistry/physics calories, or 1 kilocalorie, and is the amount of energy needed to raise 1 kilogram of water by 1°C). The calorie unit is not used as commonly in some fields today, where the joule is the standard unit of energy.