Challenge Level

Water naturally dissociates into an equilibrium mixture of $H^+$ and $OH^-$ ions and $H_2O$ molecules

$$

H_2O \rightleftharpoons^{K_W} H^++OH^-\,,

$$

where the concentrations of $H^+$ and $OH^-$ ions, written as $[H^+]$ and $[OH^-]$ are related by the expression

$$

K_W = [H^+][OH^-].

$$

$K_W$ is called the dissociation constant, and depends on the temperature of the water.

The following table of data shows the dissociation constant for water at various temperatures and standard pressure.

Water temperature | $\quad K_W\times10^{14}\quad$ |

$0^\circ$ C | 0.1 |

$10^\circ$ C | 0.3 |

$18^\circ$ C | 0.7 |

$25^\circ$ C | 1.2 |

$30^\circ$ C | 1.8 |

$50^\circ$ C | 8.0 |

$60^\circ$ C | 13 |

$70^\circ$ C | 21 |

$80^\circ$ C | 35 |

$90^\circ$ C | 53 |

$100^\circ$ C | 73 |

From this table, work out an estimate for the temperature at which water has a $pH$ of exactly 7, 6.8 and 7.2. Recall that the $pH$ is defined as $pH=-\log_{10}([H^+])$