Question

Suppose an object with initial temperature \(T_{0}\) is placed in a sealed container, which is in turn placed in a medium with temperature \(T_{m} .\) Let the initial temperature of the container be \(S_{0} .\) Assume that the temperature of the object does not affect the temperature of the container, which in turn does not affect the temperature of the medium. (These assumptions are reasonable, for example, if the object is a cup of coffee, the container is a house, and the medium is the atmosphere.) (a) Assuming that the container and the medium have distinct temperature decay constants \(k\) and \(k_{m}\) respectively, use Newton's law of cooling to find the temperatures \(S(t)\) and \(T(t)\) of the container and object at time \(t\). (b) Assuming that the container and the medium have the same temperature decay constant \(k\), use Newton's law of cooling to find the temperatures \(S(t)\) and \(T(t)\) of the container and object at time \(t\) (c) Find \(\lim _{\cdot t \rightarrow \infty} S(t)\) and \(\lim _{t \rightarrow \infty} T(t)\).

Short answer

Answer: The temperature of an object in a container with distinct temperature decay constants is given by the equation: \[T(t) = -k(T(t) - T_m - (S_0 - T_m)e^{-kt})\] When both the container and the medium have the same decay constant, the temperature of the object is given by: \[T(t) = -k_m(T(t) - T_m - (S_0 - T_m)e^{-kt})\] In both cases, solving for the exact temperature at any given time \(T(t)\) will depend on the specific values of the constants. As time approaches infinity, both the container's temperature and the object's temperature will approach the medium's temperature, \(T_m\).

Step-by-step solution

Apply Newton's law of cooling for the container

Newton's law of cooling states that the rate of change of temperature is proportional to the difference between the current and surrounding temperatures. We can write a differential equation for the container's temperature, \(S(t)\), which is influenced by the medium temperature \(T_m\): \[\frac{d S(t)}{d t} = -k(S(t) - T_m)\]

Solve the ODE for \(S(t)\)

Solve the ODE by separating variables and integrating: \[\int \frac{d S(t)}{S(t) - T_m} = -k \int dt\] Applying the integration, we get: \[\ln|S(t) - T_m | = -kt + C_1\] Solve for \(S(t)\): \[S(t) = T_m + (S_0 - T_m)e^{-kt}\] This is the temperature of the container at any time \(t\).

Apply Newton's law of cooling for the object

Now, we can write a differential equation for the object's temperature, \(T(t)\), since it is influenced by the container's temperature \(S(t)\): \[\frac{d T(t)}{d t} = -k_m(T(t) - S(t))\]

Solve the ODE for \(T(t)\) with \(S(t)\)

Replace \(S(t)\) from step 2 into the ODE from step 3 and solve for the object's temperature \(T(t)\): \[\frac{d T(t)}{d t} = -k_m(T(t) - T_m - (S_0 - T_m)e^{-kt})\]

Solve the ODE for \(T(t)\)

Solving for \(T(t)\) is not elegant and depends on the specific values of the constants. However, this equation represents the object's temperature at any time \(t\). #b) Case 2: Same decay constants for the container and medium#

Apply Newton's law of cooling for the container and object

We have the same situation as before, but now \(k_m = k\). So, we can write a differential equation for the object's temperature, \(T(t)\), influenced by the container's temperature \(S(t)\), which is influenced by the medium temperature \(T_m\): \[\frac{d T(t)}{d t} = -k(T(t) - T_m - (S_0 - T_m)e^{-kt})\]

Solve the ODE for \(T(t)\)

Same as for case (a), solving for \(T(t)\) is not simple and depends on the specific values of the constants. This equation represents the object's temperature at any time \(t\). #c) Find limits as t approaches infinity#

Find the limit of \(S(t)\) as \(t \rightarrow \infty\)

As \(t\) approaches infinity, \(e^{-kt} \rightarrow 0\), so the limit is: \[\lim_{t\to\infty} S(t) = T_m\]

Find the limit of \(T(t)\) as \(t \rightarrow \infty\)

The expression for \(T(t)\) is complex, but we do know the limit of \(S(t)\) as \(t\) approaches infinity. As \(t\) approaches infinity, the object's temperature will become closer to the container's temperature, which approaches \(T_m\). Thus, the limit is: \[\lim_{t\to\infty} T(t) = T_m\]

Key concepts

Differential Equations

Differential equations are mathematical tools that relate a function with its derivatives. They are used extensively to model situations where a rate of change is known, and the goal is to determine the function itself. For example, Newton's law of cooling, which is fundamental in thermodynamics, can be expressed as a differential equation.

In the context of our question, Newton's law is used to create two differential equations representing the rates of change of temperatures for both the container, denoted as \(S(t)\), and the object, denoted as \(T(t)\), under the influence of their environment's temperature \(T_m\). These equations allow us to explore how temperatures evolve over time given certain initial conditions and constants of proportionality that relate to the specific physics of the problem; in this case, these are the decay constants \(k\) and \(k_m\). The solution to such an ordinary differential equation (ODE) typically involves techniques such as separation of variables and integration, which provide us with functions describing the temperature over time.

Temperature Decay Constant

The temperature decay constant is a value that quantifies how quickly a substance adjusts its temperature to the surrounding environment's temperature, according to Newton's law of cooling. It is represented by \(k\) in our equations. Intuitively, the larger the decay constant, the more rapidly the temperature of the substance will converge to the environmental temperature.

The decay constant can differ between substances and environments. In the exercise, two distinct constants \(k\) and \(k_m\) are considered for the container and the medium respectively, and also a scenario where they share the same decay constant. This distinction is critical as it changes the behavior of the differential equations that model the system, thus impacting the solution and providing insight into how the decay constant affects the rate of temperature change over time. Ultimately, the decay constant plays a pivotal role in determining how the temperature functions \(S(t)\) and \(T(t)\) evolve over time.

Limit of a Function

The concept of the limit of a function is fundamental in calculus and analysis. It describes the behavior of a function as its input approaches a certain value. In the context of temperature and Newton's law of cooling, we examine the limit of the temperatures of the container and the object as time \(t\) approaches infinity, denoted as \(\lim_{t \to \infty} S(t)\) and \(\lim_{t \to \infty} T(t)\) respectively.

As the time increases, the effect of the initial temperatures diminishes due to the exponential decay factor \(e^{-kt}\), causing both \(S(t)\) and \(T(t)\) to approach the medium's steady temperature \(T_m\). Therefore, regardless of the initial temperatures and decay constants, the container and the object will eventually stabilize at the temperature of the surrounding environment. This limit helps to understand the long-term behavior of thermal systems and can be essential in engineering and environmental studies.