A mass of 1.00 kg of water at temperature T is poured from a...

A mass of $1.00 \mathrm{kg}$ of water at temperature $T$ is poured from a height of $0.100 \mathrm{km}$ into a vessel containing water of the same temperature $T,$ and a temperature change of $0.100^{\circ} \mathrm{C}$ is measured. What mass of water was in the vessel? Ignore heat flow into the vessel, the thermometer, and so on.

Key Concepts

Gravitational Potential Energy

This concept refers to the energy possessed by an object due to its position in a gravitational field. In problems like this, the water falling from a height acquires kinetic energy by converting gravitational potential energy (calculated as mgh, where m is mass, g is the acceleration due to gravity, and h is the height). This energy is the starting point for later energy transformations that affect the thermal state of the water.

Energy Conversion

Energy conversion involves the transformation of energy from one form to another, in this case from gravitational potential energy to kinetic energy and ultimately to thermal energy. When the falling water impacts and mixes, its mechanical energy is dissipated into heat, causing a measurable increase in temperature. Recognizing this conversion is crucial for setting up the energy balance used to calculate the amount of water in the vessel.

Specific Heat Capacity

Specific heat capacity is a property that quantifies the amount of heat needed to change a unit mass of a substance by one degree in temperature. For water, which has a high specific heat capacity, the presence of even a small amount of energy can result in only a slight temperature increase. This concept is key in linking the absorbed heat (from energy conversion) to the resultant temperature change observed in the water.

Conservation of Energy in Thermal Processes

This principle states that energy can neither be created nor destroyed, only converted from one form to another. In the scenario presented, the mechanical energy lost by the falling water is completely converted into thermal energy that raises the temperature of the water in the vessel. Applying energy conservation allows us to equate the gravitational potential energy of the falling water to the thermal energy gain in the combined water mass, thereby enabling the calculation of the mass of water in the vessel.

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