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Water permittivity at temperature

What is the dielectric constant of the water in the cell once it has warmed up?

The formula
ε=e·1−0.004⁢T−20
ε
Relative permittivity at T
e
Relative permittivity at 20 °C
T
Temperature, °C
LaTeX 
ε = e \left( 1 - 0.004\,(T - 20) \right)

Work it out

Fill from a part the library

What is in the cell. Sets the permittivity at 20 °C and the conductivity.

°C

The temperature of the water, not of the room. A cell under load runs warmer than its surroundings.

Compare with a variation

Result

ε Relative permittivity 80.1

The dielectric constant at this temperature. Multiply the cell’s vacuum capacitance by this.

Δ Change from 20 °C 0

How far the permittivity has moved, as a fraction. Capacitance moves with it, and resonance by about half as much.

With your numbers
80.1=80.1·1−0.004⁢20 °C−20
LaTeX 
80.1 = 80.1 \left( 1 - 0.004\,(20\,\mathrm{°C} - 20) \right)

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Relative permittivity against temperature Temperature swept from 10 °C to 30 °C with everything else held at your numbers. The dashed lines cross where you are.
Relative permittivity against temperatureRelative permittivity falls from 83.3 to 76.9 as temperature rises from 10 °C to 30 °C. At your temperature of 20 °C it is 80.1.7678808284101520253020 °C80.1Temperature (°C)Relative permittivity
The formula behind the curve
ε=e·1−0.004⁢T−20
ε
Relative permittivity at T
e
Relative permittivity at 20 °C
T
Temperature, °C
LaTeX 
ε = e \left( 1 - 0.004\,(T - 20) \right)
What moves the answer Each input moved 10% either way, with the others held still, and the effect on relative permittivity.
What moves the answerRelative permittivity is most sensitive to Temperature, which moves it by about 0.8% for a 10% change. It is the only input.Change in the answer when each input moves by 10%-1%-0.5%0%0.5%1%Temperature±0.8
The formula behind the curve
ε=e·1−0.004⁢T−20
ε
Relative permittivity at T
e
Relative permittivity at 20 °C
T
Temperature, °C
LaTeX 
ε = e \left( 1 - 0.004\,(T - 20) \right)

Method

  1. Take the water’s relative permittivity at 20 °C — 80.1 for distilled, 78.5 for tap, 76 for mineral, 74 for sea water.
  2. Find how far the temperature is from 20 °C. Above it the permittivity falls; below it, rises.
  3. Water loses about 0.4 % of its permittivity per degree of warming. Multiply the difference by 0.004 and subtract that fraction.
  4. The cell’s capacitance is directly proportional to this, so the same fraction moves the capacitance. Resonant frequency goes as one over the square root of capacitance, so it moves about half as far, and the other way.

Assumptions

  • The 0.4 % per degree coefficient is linear. It is a good approximation between roughly 0 °C and 60 °C and increasingly poor outside that; near boiling the real curve is steeper than this straight line.
  • The water is at one temperature throughout. A cell under load is warmer at the plates than at the wall, and what matters for capacitance is the average across the gap.
  • This is the static (low-frequency) permittivity. Above about a gigahertz water’s permittivity falls away regardless of temperature — see the Cole-Cole calculation for that.
  • Dissolved solids are not modelled here. They matter much more for conductivity and loss than for permittivity, which is why the four profiles differ by a few per cent in ε and by six orders of magnitude in σ.