A "clever" technician decides to warm some water for his...

A "clever" technician decides to warm some water for his coffee with an x-ray machine. If the machine produces 10.0 $\mathrm{rad} / \mathrm{s}$ , over what time interval will the temperature of an insulated cup of water rise by $50.0^{\circ} \mathrm{C} ?$

Key Concepts

Time Interval Calculation in Energy Transfer

Calculating the time required for a given temperature change involves equating the energy absorbed (which is a product of the radiation dose, its conversion to energy, and the mass of the material) to the energy required to heat the material (given by mass times specific heat capacity times the temperature change). This relationship allows one to solve for the time interval during which energy is deposited until the desired temperature increase is reached.

Specific Heat Capacity

The specific heat capacity is a material property that indicates the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this number is approximately 4186 J/(kg·°C). This concept is crucial when determining how much energy is needed to achieve a specific temperature change, linking the absorbed energy from radiation to the resulting temperature increase.

Energy Deposition and Absorption

When radiation interacts with matter, it deposits energy, which can increase the thermal energy of the material. This concept connects the dose rate provided by a device (given in rad/s) to the actual energy absorbed per unit mass over time. This energy deposition is the bridge between the provided radiation and the heating of the water.

Radiation Dose Units

The rad is a unit of absorbed radiation dose that quantifies the amount of energy deposited per unit mass in a material. It is defined such that 1 rad corresponds to 0.01 joules of energy absorbed per kilogram of substance. Understanding this unit is key to relating the radiation output of devices like x?ray machines to the energy absorbed by matter, which is essential for calculating subsequent temperature changes.

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