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An equilibrium statistical ensemble with fixed particle number, pressure and temperature, allowing volume and energy to fluctuate. With a temperature-independent volume measure, its joint weight is proportional to $e^{-\beta (E+PV)}$. Its normalization and any temperature-dependent prefactor must be specified when taking thermodynamic derivatives.

A steady spherical Parker wind with constant isothermal sound speed $c$ and negligible wind self-gravity obeys $4\pi r^2\rho u=\dot{M}$ and $(u-c^2/u)u^{\prime }=2c^2/r-GM/r^2$. A regular sonic point has $u=c$ at $r_s=GM/(2c^2)$. With $x=r/r_s$ and $w=u^2/c^2$, integration gives the displayed equation for the smooth accelerating transonic branch. Its slope at the sonic point is $u^{\prime }(r_s)=c/r_s$. Maintaining constant gas temperature during expansion generally requires heat input, so this model is distinct from an adiabatic wind with $\gamma >1$.

If an isothermal thin atmosphere has surface radius $R_p$, surface pressure $P_0$, and nearly constant gravity, the radius of a pressure level $P$ is

For fixed mass and temperature, $p\propto L^{-3}$, so the pressure-support scale $p{L}^3$ stays constant while gravitational binding scales as $L^{-1}$. Isothermal pressure therefore becomes relatively less important during homologous contraction.

An isothermal process changes a thermodynamic system while keeping its temperature constant. It need not keep its pressure, volume or internal energy constant, and it does not generally have zero heat transfer. An adiabatic process specifies zero heat transfer instead. For example, a slowly expanding ideal gas can absorb heat from a heat bath and deliver the same amount of work while its temperature stays fixed.

For a fixed amount of substance and composition, internal energy of an ideal gas depends only on temperature. With $Q>0$ for heat entering and $W_{\mathrm{by}}>0$ for work delivered to the surroundings, the first law of thermodynamics therefore givesThis equality does not determine either transfer until the path is specified. Isothermal ideal-gas work gives the reversible value; an insulated free expansion has a different value even when its equilibrium endpoints have the same temperature. The closed system convention here excludes changes in bulk kinetic energy and external potential energy.

A conducting cylinder in a large, stirred heat bath approximates an isothermal process when the piston moves slowly enough for thermal conduction to keep up. A quasistatic process keeps the gas close to internal thermodynamic equilibrium; a reversible thermodynamic process additionally requires negligible friction and vanishing driving differences in pressure and temperature. Exact equality of the gas and heat bath temperatures gives no finite heat transfer rate through a finite thermal resistance. The reversible description is a limit of arbitrarily small differences and arbitrarily slow motion, illustrated by the finite-rate isothermal piston model.^[ref]

For a real gas, internal energy at constant temperature can change with volume because molecular interaction potential energy changes. In one smooth thermodynamic phase of a fixed-composition simple compressible system,Thus an isothermal process generally has $Q=\Delta U+W_{\mathrm{by}}$, with no reason to set $\Delta U=0$. The right-hand side vanishes for the ideal gas law, but not generally for another equation of state.^[ref]

An isothermal shock joins two states at the same temperature and isothermal sound speed. Mass and momentum conservation imply $u_1{u}_2=c_s^2$ for a nontrivial planar jump, so the compression ratio is the upstream isothermal Mach number squared. Heat exchange replaces adiabatic energy conservation across the gas alone.

Without an escape-energy cutoff, a galactic distribution function proportional to $\exp [(\Psi -v^2/2)/{\sigma }^2]$ gives $\rho \propto e^{\Psi /{\sigma }^2}$ and a constant one-dimensional velocity dispersion. Since $\Psi ={\Phi }_0-\Phi$, differentiation gives the displayed hydrostatic relation. An ideal gas at constant temperature has pressure $P=\rho k_{B}T/m_0$, so its hydrostatic equilibrium equation is identical at the displayed dispersion-temperature identification. Coupling either relation to the same Poisson equation for Newtonian gravity, with matched central density and boundary data, therefore gives the same density structure. Truncating the stellar velocities at escape changes this equivalence.

An isothermal terminal block keeps the two reference transitions and the reference thermometer at approximately the same temperature, while maintaining electrical insulation between the terminals. Thermal contact is required; an electrical short between A and B would bypass the signal. Merely using copper does not eliminate the Seebeck effect in the meter leads. If both homogeneous copper leads start at $T_c$ and end at a common meter-input temperature $T_m$, their contributions have equal magnitude and opposite sign around the measurement path:This cancellation uses matched endpoint temperatures and homogeneous lead material, rather than $S_{\mathrm{Cu}}=0$. A temperature gradient along each lead is allowed under these assumptions. If the two transitions are at different temperatures, one measured $T_c$ cannot generally correct both; unmatched connectors or wire inhomogeneity can introduce additional electromotive forces. The FP-TC-120 documentation explicitly locates the reference sensing in the terminal base or connector block and specifies isothermal versions of these assemblies.^[ref]

Isothermal titration calorimetry measures heat accompanying additions of one binding partner to another at controlled temperature. With suitable concentration ranges, stoichiometry and a binding model, it can estimate affinity and binding enthalpy; dilution and buffer effects need controls.

A physicist.^[ref][ref]

A physicist who developed semiconductor lasers at Bell Laboratories.^[ref]

James Prescott Joule was an English physicist whose private experiments connected electricity, heat and mechanical work. His family ran a brewery in Salford; he developed an experimental laboratory while working in the family business.^[ref] The Royal Society's archive identifies his occupation as physicist and records his life as 1818–1889.^[ref]

His electrical heating experiments placed metal coils in water and measured the temperature change. Later, he used the brewery cellar to reduce ambient temperature fluctuations during friction experiments and took particular care over thermometer calibration.^[ref] These practical choices address a central difficulty of calorimetry: a small temperature rise can reflect both the intended input of energy and unwanted heat transfer to the surroundings. A quiet baseline and a known heat capacity are part of the measurement, not decoration around the formula.

The contrast between Joule heating and Joule's paddle-wheel experiment makes the connection especially clear. One supplies electrical work; the other supplies mechanical work. In each case, calorimetry tracks the resulting change of internal energy. Comparing quantitatively controlled inputs with thermal changes gives an experimental test of conservation of energy, rather than merely showing that stirring or an electric current can warm water. The paddle-wheel account describes the descending weights, vanes, thermal controls and corrections needed for that comparison.

Joule's free-expansion experiment asks a different question: how does a gas's temperature change during expansion without external work? The Joule-Thomson effect instead concerns flow through a restriction and a balance involving enthalpy. Keeping the two experiments distinct prevents a shared historical name from hiding different system boundaries and different conserved quantities.

A physicist associated with optical rotation and Biot's saccharimeter.

For particles of mass $m$ at radius $r$ and temperature $T$, the Jeans escape parameter isLarge $\lambda$ gives weak tail escape, while order-unity $\lambda$ indicates hydrodynamic blow-off.

Astronomer and author of research on X-ray plasma spectroscopy.