The isothermal surfaces are coaxial cylinders.
The isothermal lines run almost due E.
2, let BC be a small portion of any isothermal corresponding to the temperature 0', and AD a neighbouring isothermal 0".
It is often impossible to observe the pressure-coefficient dp/de directly, but it may be deduced from the isothermal compressibility by means of the geometrically obvious relation, BE = (BEÃ†C) XEC. The ratio BEÃ†C of the diminution of pressure to the increase of volume at constant temperature, or - dp/dv, is readily observed.
I by the whole area B"DZ'VO under the isothermal 9"D and the adiabatic DZ', bounded by the axes of pressure and volume.
The difference 90-E is represented by the area 9"DdO to the left of the isometric Dd under the isothermal B"D.
And 33° S., or approximately within the isothermal lines of 60° F.
A cycle such as ABCD enclosed by parts of two isothermals, BC, AD, and two adiabatics, AB, CD, is the simplest form of cycle for theoretical purposes, since all the heat absorbed, H', is taken in during the process represented by one isothermal at the temperature o', and all the heat rejected, H", is given out during the process represented by the other at the temperature 0".
Taking this ideal limit as a theoretical or absolute zero, the value of H may be represented on the diagram by the whole area included between the two adiabatics BAZ, CDZ' down to the points where they intersect the isothermal of absolute zero, or the zero isopiestic OV asymptotically at infinity.
Applying the above equation to a gas obeying the law pv=RT, for which the work done in isothermal expansion from a volume i to a volume r is W=RT loger, whence dW=R log e rdt, he deduced the expression for the heat absorbed by a gas in isothermal expansion H=R log er/F'(t).
Done by a gas in isothermal expansion is assumed to be equivalent or proportional to the heat absorbed, H=R log e r/F'(t).
This most fundamental point was finally settled by a more delicate test, devised by Lord Kelvin, and carried out in conjunction with Joule (1854), which showed that the fundamental assumption W =H in isothermal expansion was very nearly true for permanent gases, and that F'(t) must therefore vary very nearly as J/T.
Let BE be an isometric through B meeting AD in E, and EC an isopiestic through E meeting BC in C. Let BA, CD be adiabatics through B and C meeting the isothermal 0" in A and D.
Then by relations (2) the heat, H, absorbed in the isothermal change BC, is to the work, W, done in the cycle ABCD in the ratio of o to (o' - o").
EF is the change of volume corresponding to a change of pressure BE when no heat is allowed to escape and the path is the adiabatic BF, EC is the change of volume for the same change of pressure BE when the path is the isothermal BC. These changes of volume are directly as the compressibilities, or inversely as the elasticities.
If we write K for the adiabatic elasticity, and k for the isothermal elasticity, we obtain S/s = ECÃ†F = K/k.
2, may be found by adding to the heat required for the change of temperature at constant volume, sdo, or at constant pressure, Sdo, the heat absorbed in isothermal expansion as given by relations (4).
It is generally convenient to divide the path into two steps, isothermal and isometric, or isothermal and isopiestic, and to integrate along each separately.
The isothermal elasticity - v(dp/dv) is equal to the pressure p. The adiabatic elasticity is equal to y p, where -y is the ratio S/s of the specific heats.
Joule failed to observe any change of temperature in his apparatus, and was therefore justified in assuming that the increase of intrinsic energy of a gas in isothermal expansion was very small, and that the absorption of heat observed in a similar experiment in which the gas was allowed to do external work by expanding against the atmospheric pressure was equivalent to the external work done.
The increment of this area (or the decrement of the negative area E--04) at constant temperature represents the external work obtainable from the substance in isothermal expansion, in the same way that the decrement of the intrinsic energy represents the work done in adiabatic expansion.
The function G is represented by the negative area D"DM under the isothermal, bounded by the isopiestic DM and the axis of pressure.
K, k, Adiabatic and isothermal elasticities.
The isothermal lines trend from south-east to north-west.
The isothermal lines, in fact, suggest that in the vast area of the Pacific something corresponding to the " planetary circulation " is established, further investigation of which may be of extreme value in relation to current inquiries concerning the upper air.
A continual circulation might thus be set up in an isothermal enclosure and maintained with the performance of an unlimited supply of work.
In considering the corresponding relation for a solution instead of a pure liquid, possible differences in concentration make the column method difficult of application, and it is better to attach the problem by means of an imaginary cycle of isothermal operation.
The available energy A is the work which may be gained from the system by a small reversible isothermal operation with an osmotic cylinder, that is Pdv.
(I) This relation applies accurately to the case of the steady flow of heat in parallel straight lines through a homogeneous and isotropic solid, the isothermal surfaces, or surfaces of equal temperature, being planes perpendicular to the lines of flow.
The Peculiar Advantage Of The Electric Method Of Callendar And Barnes, Already Referred To, Is That The Specific Heat Itself Is Determined Over A Range Of 8° To 10° At Each Point, By Adding Accurately Measured Quantities Of Heat To The Water At The Desired Temperature In An Isothermal Enclosure, Under Perfectly Steady Conditions, Without Any Possibility Of Evaporation Or Loss Of Heat In Transference.
The stages at which heat is taken from the furnace and rejected to the cooler (C) are approximately isothermal at the upper and lower limits of temperature respectively, and the cycle accordingly is approximately "perfect" in the thermodynamic sense.
The theoretical indicator diagram is made up of two isothermal lines for the taking in and rejection of heat, and two lines of constant volume for the two passages through the regenerator.
It was then discharged through the regenerator, depositing heat for the next charge of air in turn to take up. The indicator diagram approximated to a form made up of two isothermal lines and two lines of constant pressure.
These isothermal lines will be found to vary frommonth to month over the two hemispheres, or over local areas, during summer and winter, and their position is modified by continental or oceanic conditions.
Thus, the tiger ranges from the equator to northern Asia as far as the river Amur, and to the isothermal of 32° Fahr.
The temperature, however, has a daily range less than that of other countries under the same isothermal lines.
(I) The heat, H, absorbed in isothermal expansion (latent heat of expansion) from p to p" is equal to the diminution of pressure (p' - p”) multiplied by the absolute temperature and by the expansion per degree (v" - v')/(o' - o") at constant pressure.
(2) The heat, H, absorbed in isothermal expansion from v to v" is equal to the increase of volume (v" - v') multiplied by the absolute temperature, and by the increase of pressure per degree (p' - p")/(o' - o"), at constant volume.
The heat absorbed in isothermal expansion from vo to v at a temperature 0 is equal to the work done by equation (8) (since d0 =o, and 0(dp/d0)dv =pdv), and both are given by the expression RO log e (v/vo).
I) to any other adiabatic 0", the quotient H/o of the heat absorbed by the temperature at which it is absorbed is the same for the same two adiabatics whatever the temperature of the isothermal path.
In virtue of relations (2), the change of entropy of a substance between any two states depends only on the initial and final states, and may be reckoned along any reversible path, not necessarily isothermal, by dividing each small increment of heat, dH, by the temperature, 0, at which it is acquired, and taking the sum or integral of the quotients, dH/o, so obtained.
In the case of a solid or a liquid, the latent heat of isothermal expansion may often be neglected, and if the specific heat, s, be also taken as constant, we have simply 0-00 =s log e0/00.
In the special case, however, in which the transformation is conducted in an isothermal enclosure, a common condition easily realized in practice, the temperature at the end of the transformation is reduced to its initial value throughout the substance.