Question

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its own temperature and the ambient temperature. What is the mathematical expression for Newton's law of cooling?

1) T(t) = T_a + (T_0 - T_a)e⁽⁻ᵏᵗ⁾
2) T(t) = T_0 + (T_a - T_0)e⁽⁻ᵏᵗ⁾
3) T(t) = T_0 + (T_a - T_0)kt
4) T(t) = T_a + (T_0 - T_a)kt

Answer 1

The correct mathematical expression for Newton's law of cooling is T(t) = T_a + (T_0 - T_a)e^(-kt), where T(t) is the temperature at time t, T_0 is the initial temperature, T_a is the ambient temperature, and k is a constant.

Step-by-step explanation:

The mathematical expression for Newton's law of cooling is that the temperature of an object changes at a rate proportional to the difference between its own temperature and the ambient temperature. The correct expression from the given options is T(t) = T_a + (T_0 - T_a)e^(-kt). Here, T(t) represents the temperature of the object at time t, T_0 is the initial temperature of the object, T_a is the ambient temperature, and k is the positive constant of proportionality depending on the characteristics of the object and the environment.

Newton's law of cooling is utilized in various scientific and engineering applications, such as predicting the cooling rate of materials and designing cooling systems. It is important to note that this law is a good approximation only under certain conditions, such as when the temperature difference is relatively small, and the heat transfer is dominated by convection.