The surface tension of water \(\gamma\) in dynes per centimeter is given by the empirical equation (W. D. Harkins: The Physical Chemistry of Surface Films, Reinhold, New York, 1952, p. 76.), $$ \gamma=75.796-0.145 \theta-0.00024 \theta^{2} $$ where \(\theta\) is the Celsius temperature. Caiculate the change of surface tension with respect to temperature \(\theta, d \gamma / d \theta\), at \(10^{\circ} \mathrm{C}\) and \(60^{\circ} \mathrm{C}\).

An empirical equation is a formula derived from experimental data rather than theory. It's often used to summarize and predict outcomes based on observed trends.
In this case, the empirical equation for the surface tension of water is given as: \[ \gamma = 75.796 - 0.145 \theta - 0.00024 \theta^2 \]Here, \theta represents the temperature in Celsius. These types of equations can be very useful for making quick predictions about complex phenomena without needing an in-depth scientific explanation. The constants (75.796, -0.145, and -0.00024) are derived from data collected through experiments on water's surface tension at different temperatures.

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function changes at any point. In simpler terms, it tells you how a quantity like surface tension changes as temperature changes.
In our problem, we start with the empirical equation\[\gamma = 75.796 - 0.145 \theta - 0.00024 \theta^2\]To find how surface tension changes with temperature, we differentiate with respect to \theta:\[ \frac{d \gamma}{d \theta} = -0.145 - 0.00048 \theta \]This new equation gives us the rate of change of surface tension \gamma at any temperature \theta.
By substituting the value of \theta in this differentiated equation, we can find out how much the surface tension changes at any specific temperature.

Surface tension is sensitive to temperature changes. As temperature increases, the liquid molecules move more vigorously and spread out more, so the surface tension usually decreases.
In our equation, the temperature \theta affects the surface tension \gamma directly:\[\frac{d \gamma}{d \theta} = -0.145 - 0.00048 \theta\]
For example, at 10°C:\[ \frac{d \gamma}{d \theta} = -0.145 - 0.00048 \times 10 \approx -0.1498 \]And at 60°C:\[ \frac{d \gamma}{d \theta} = -0.145 - 0.00048 \times 60 \approx -0.1738 \]
The calculations show that surface tension drops more rapidly at higher temperatures, highlighting the importance of understanding this relationship in practical applications, such as in chemical processes and industrial applications.