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While you are studying for an upcoming physics exam, a lightning storm is brewing outside your window. Suddenly, you see a tree across the street struck by lightning. There is a loud sound and you see smoke rising from various parts of the tree. You stop studying for your exam and do online research on lightning, trees, and sap. You find that a typical lightning bolt represents a potential difference of 1.00 ✕ 108 V between the cloud and the ground and that it can transfer a typical charge of 50.0 C between the cloud and the ground. Some models show that when a tree is hit by lightning, perhaps 2.00% of the energy in the lightning bolt can be delivered to the sap, causing it to boil. Model the sap as water initially at 30.0°C. If all of the sap in the tree is vaporized to gaseous sap at 100°C, determine how much sap (in kg) there is in the tree. (Use any necessary values found in this table or this table.)

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The mass of sap in the tree is 96.2 kg

Electrical energy between the cloud and ground

The electrical energy between the cloud and ground is the energy stored in a capacitor W = 1/2QV where

  • Q = charge between cloud and ground = 50.0 C and
  • V = potential difference between cloud and ground = 1.00 × 10⁸ V.

So, W = 1/2QV

= 1/2 × 50.0 C × 1.00 × 10⁸ V

= 25.0 × 10⁸ J

Energy used to boil sap

The energy used to boil the sap from 30 °C to 100 °C and modelling the sap as water, is

E = mc(T' - T) + mL where

  • m = mass of sap,
  • c = specific heat capacity of sap = 4200 J/kg-°C,
  • T = initial temperature of sap = 30 °C,
  • T' = final temperature of sap = 100 °C and
  • L = latent heat of vaporization of sap = 226,000 J/kg

Mass of sap

Making m subject of the formula, we have

m = E/[c(T' - T) + L]

Now since energy the lightning bolt delivers to the sap is 2.00 %. So, E = 2.00 % W, we have that E = 0.02W

So, m = 0.02W/[c(T' - T) + L]

Substituting the values of the variables into the equation, we have

m = 0.02W/[c(T' - T) + L]

m = 0.02 × 25.0 × 10⁸ J/[4200 J/kg-°C(100 °C - 30 °C) + 226,000 J/kg]

m = 0.5.0 × 10⁸ J/[4200 J/kg-°C(70°C) + 226,000 J/kg]

m = 0.5 × 10⁸ J/[294000 J/kg + 226,000 J/kg]

m = 5 × 10⁷ J/520000 J/kg

m = 50 × 10⁶ J/0.52 × 10⁶ J/kg

m = 96.15 kg

m ≅ 96.2 kg

The mass of sap in the tree is 96.2 kg

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