Specific Heat. 155. It is found that a certain quantity of heat must be expended in order to raise the temperature of a mass of any substance by a given amount. The requisite quantity of heat depends on the nature of the substance and also on its mass, and for. any particular substance it may be at once assumed that the quantity of heat required to raise the temperature one degree is directly proportional to the mass of the substance. In general, the amount of heat required to change the temperature of a given mass from to to (t+1)o is the same for all values of t. Hence for the same substance the quantity of heat expended in changing the temperature from to to to att when the mass is given, and the mass when t'-t is given, and therefore generally a m(t-t), if m be the mass. If this be taken equal to cm (t'-t), c is called the specific heat of the substance, and it is the measure of the amount of heat which will raise by 1° the temperature of the unit of mass. If two masses m, m', of the same substance, at temperatures t, t, be mixed together, and if be the temperature of the mixture, then, since the amount of heat lost by one is gained by the other, mt-mr=m'T-Wi't, or mt + m't' = (m+m') T. 156. For different substances the quantity c has different values; thus it is found that water requires about 28 times as much heat as mercury in order to change the temperature by a given amount, and the specific heat of mercury is therefore less than that of water in the ratio of 1: 28. The specific heat of a gas must be considered from two different points of view, for we may suppose the volume of a gas constant, and investigate the amount of heat required to raise the temperature 1o, or we may suppose the pressure constant, the latter supposition permitting the expansion of the gas. The specific heat in the second case exceeds the specific heat in the first case by the amount of heat disengaged when the gas is suddenly compressed into its original volume. 157. The specific heat of water is usually taken as the unit, and one of the methods of finding, the specific heat of a substance is by immersing it in a given weight of water, and observing the temperature attained by the two substances. Thus, if M be the mass of a body, Tits temperature, and C its specific heat, m' and m the masses of a vessel and of the water in it, and t their common temperature, Tthe temperature of the whole after immersion, and C the specific heat of the vessel, CM (T-7)=m (r− t) + C'm' (T − t), since the quantity of heat lost by the body is equal to that gained by the water and the vessel. If C be known, this equation determines C; and C', if unknown, can be found by pouring water of a known temperature into the vessel at some other known temperature. The following are approximate values of the specific heats of a few substances. EXAMINATION ON CHAPTER VIII. 1. A CUBIC foot of air having a pressure of 15 lbs. on a square inch is mixed with a cubic inch of compressed air, having a pressure of 60 lbs. on a square inch; find the pressure of the mixture, when its volume is 1729 cubic inches. 2. State the conditions under which a space is saturated with vapour. 3. A vessel of water is left in a close room for some time; what would be the effect of bringing a quantity of ice into the room? 4. Explain the radiation, conduction, and convection of heat. Why is a cloudy sky not favourable to the deposition of dew? 5. How do you account for the long trail of condensed steam which often follows a locomotive in rainy weather? 6. Define the Dew-point, and explain the use of the Wet and Dry Bulb Thermometer. 7. Explain why it is difficult to heat water from its upper surface. 8. If a piece of ice be put into a glass of water, the external surface is soon covered with a fine dew; account for this fact. 9. Explain what is meant by Specific Heat. Three gallons of water at 45° are mixed with six gallons at 90°; what is the temperature of the mixture? 10. At great altitudes it is sometimes found that a sensation of discomfort is felt; the lips crack and the skin of the hands is roughened; how do you account for these facts? Can you give any reason why an east wind in England sometimes produces similar effects? EXAMPLES. 1. Two volumes V, V' of different gases, at pressures p, p', and temperature t are mixed together; the volume of the mixture is U, and its temperature t', determine the pressure. 2. Two vessels contain air having the same temperature t, but different pressures p, p'; the temperature of each being increased by the same quantity, find which has its pressure most increased. If the vessels be of the same size, and be allowed to communicate with each other, find the pressure of the mixture at a temperature zero. 3. A glass vessel weighing 1 lb. contains 5 oz. of water, both at 20o, and 2 oz. of iron at 100° is immersed; what is the temperature of the whole, taking .2 as the specific heat of glass and .12 of iron? 4. An ounce of iron at 120o, and 2 oz. of zinc at 90¢ are thrown into 6 oz. of water at 10° contained in a glass vessel weighing 10 oz.; what is the final temperature, taking .1 and .12 as the specific heats of zinc and iron? B. E. H. 10 5. The pressure of a quantity of air, saturated with vapour, is observed; the mixture is then compressed into half its former volume, and, after the temperature has been lowered until it becomes the same as at first, the pressure is again observed; hence find what would be the pressure of the air (occupying its original space) if it were deprived of its vapour without having its temperature changed. 6. It is related of a place in Norway that a window of a ball-room being suddenly thrown open, a shower of snow immediately fell over the whole of the room. Account for this phe nomenon. 7. A drop of water is introduced into the tube of a common barometer which just does not evaporate at the higher of the temperatures t1o, tëo. Given that the elasticity of vapour increases geometrically as the temperature increases arithmetically, shew that if E1, Eg be the errors of the above barometer at temperatures to, too, the common ratio of the geometric progression for an increase of temperature of 1o in the case of vapour of water is e being the coefficient of expansion for mercury. 8. A closed cylinder contains a piston moveable by means of a rod passing through an air-tight collar at the top of the cylinder. The piston is held at a distance from the bottom of the cylinder equal to one-third of its height, and vapour is introduced above and below of a known pressure, the temperature of the cylinder being such as will support vapour of twice the density without condensation. The piston on being left to itself sinks through two-ninths of the height of the cylinder. Prove that the weight of the piston is five-fourths of the pressure of the vapour upon either side at first. 9. A flask is partially filled with water which is caused to boil until the air is expelled, and then the flask is corked and allowed for a short time to cool. The flask is then placed in cold water, and it is found that the water in it recommences boiling. Explain this phenomenon. CHAPTER IX. Rotating Liquid. 158. WHEN liquid in a vessel is set rotating, it is known that the surface assumes a hollow form; by the help of a dynamical law we can determine what this form is. If a liquid, contained in a vessel which rotates uniformly about a vertical axis, rotate uniformly with the vessel, its surface is a paraboloid. Every particle of the liquid moves uniformly in a horizontal circle, and therefore whatever the forces may be which act on any particle, their resultant must be a horizontal force tending to the centre of the circle and equal to mor, where r is the distance of the particle from the axis, m its mass, and w the angular velocity of the liquid*. Let AG be the axis of rotation, and consider a particle P in the surface. The forces which act on P are its weight mg downwards, and the resultant pressure of the surrounding liquid. This resultant pressure must be normal to the surface, or else the particle would not maintain its position on the surface. Let the normal at P meet the axis in G, and draw PN horizontal. Then two forces acting in directions of the lines PG, GN have their resultant, N A which is mo2PN, in the direction PN, and therefore by the triangle of forces, NG: PN:: mg : mw2PN, *See Goodwin's Dynamics, p. 119, or Parkinson's Mechanics, Art. 105. |