Page:Elementary Text-book of Physics (Anthony, 1897).djvu/248

234 to pass across the space between the two walls and back is $$\frac{2s}{u};$$ and the number of impacts upon the first surface in unit time is $$\frac{u}{2s}\cdot$$

Consider the molecules contained in a rectangular prism, with bases of area $$a$$ in the walls. These molecules must be considered as moving in all directions and with various velocities. But the velocity of any molecule miiy be resolved in the direction of three rectangular axes, one normal to the surface and the other two parallel to it, and the effect upon the walls will be due only to the normal components. Let us single out for examination a group of molecules which have a normal velocity that lies near the value $$u_{1},$$ and let $$n_{1}$$ represent the number of such molecules in unit volume. Then the number of such molecules within the prism considered is $$n_{1}sa.$$ The number of impacts made by them in unit time on one the walls is $$n_{1}sa. \frac{u_{1}}{2s} = \frac{n_{1}au_{1}}{2},$$ and in the time $$\theta$$ is $$\frac{n_{1}au_{1}\theta}{2}\cdot$$ Hence the total pressure which they exert on the area $$a$$ is $$m\frac{2u_{1}}{\theta}. \frac{n_{1}au_{1}\theta}{2} = mn_{1}u_{1}^2a,$$ and on unit area is $$mn_{1}u_{1}^2.$$

Now the total pressure on unit of area is the sum of the pressures due to all the i groups into which the molecules of the gas may be divided, or $$p = m(n_{1}u_{1}^2 + n_{2}u_{2}^2 + . . . n_{i}u_{i}^2).$$ If we represent by $$n$$ the number of molecules in unit volume and by $$u$$ the mean velocity given by $$nu^2 = n_{1}u_{1}^2 + n_{2}u_{2}^2 +. . . n_{i}u_{i}^2,$$ we have $$p = mnu^2.$$ Similar expressions hold for the pressures on the other walls, the velocities normal to which are $$v$$ and $$w,$$ and we assume that these mean velocities are independent of direction, so that $$u^2 = v^2 = w^2 .$$ But the velocity of any molecule is given by $$V_{i}^2 = u_{i}^2 + v_{i}^2 + w_{i}^2,$$ and the mean velocity is given by a similar equation. Hence $$V^2 = 3u^2$$ and we have finally, The velocity $$V$$ in this expression is called the velocity of mean square.

If we now suppose the volume of the gas to change so that the