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I would like to know what does a gr mean when used to represent the physical state of the carbon, for example, in the synthesis of the glucose

$$\ce{6C_{(gr)} + 6H2_{(g)} + 3O2_{(g)}-> C6H12O6_{(\ell)}} \quad,$$

what does the $\mathrm{gr}$ mean?

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The letters in brackets indicate state of aggregation (includes state of matter, crystal structure (solids) and atomicity (gases)), therefore $\ce{C_{(gr)}}$ most likely refers to graphite. Often one can also see $\ce{C(s, gr)}$ or $\ce{C(s, graphite)}$ There is a nice demonstration in [1, p. 248]:

$$\ce{C(gr) + O2(g) <=> CO2(g)}$$ Here $\ce{(g)}$ indicates that the compound is gas, [...], and $\ce{(gr)}$ that carbon is graphite (not many people will burn diamonds, even if they are chemists).

P. S. According to IUPAC recommendations, subscript notation is deprecated.

  1. Metiu, H. Physical Chemistry: Thermodynamics; Garland Science, 2006. ISBN 978-1-315-31491-4.
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